diff --git a/Makefile.in b/Makefile.in
index 043fb445fa8..8195f97bd1b 100644
--- a/Makefile.in
+++ b/Makefile.in
@@ -346,16 +346,17 @@ am__libmesh_dbg_la_SOURCES_DIST = src/base/default_coupling.C \
 	src/geom/cell_tet4.C src/geom/edge.C src/geom/edge_edge2.C \
 	src/geom/edge_edge3.C src/geom/edge_edge4.C \
 	src/geom/edge_inf_edge2.C src/geom/elem.C \
-	src/geom/elem_cutter.C src/geom/elem_quality.C \
-	src/geom/elem_refinement.C src/geom/elem_side_builder.C \
-	src/geom/face.C src/geom/face_inf_quad.C \
-	src/geom/face_inf_quad4.C src/geom/face_inf_quad6.C \
-	src/geom/face_quad.C src/geom/face_quad4.C \
-	src/geom/face_quad4_shell.C src/geom/face_quad8.C \
-	src/geom/face_quad8_shell.C src/geom/face_quad9.C \
-	src/geom/face_tri.C src/geom/face_tri3.C \
+	src/geom/elem_corner.C src/geom/elem_cutter.C \
+	src/geom/elem_quality.C src/geom/elem_refinement.C \
+	src/geom/elem_side_builder.C src/geom/face.C \
+	src/geom/face_inf_quad.C src/geom/face_inf_quad4.C \
+	src/geom/face_inf_quad6.C src/geom/face_quad.C \
+	src/geom/face_quad4.C src/geom/face_quad4_shell.C \
+	src/geom/face_quad8.C src/geom/face_quad8_shell.C \
+	src/geom/face_quad9.C src/geom/face_tri.C src/geom/face_tri3.C \
 	src/geom/face_tri3_shell.C src/geom/face_tri3_subdivision.C \
-	src/geom/face_tri6.C src/geom/face_tri7.C src/geom/node.C \
+	src/geom/face_tri6.C src/geom/face_tri7.C \
+	src/geom/intersection_tools.C src/geom/node.C \
 	src/geom/node_elem.C src/geom/plane.C src/geom/point.C \
 	src/geom/reference_elem.C src/geom/reference_elem.data \
 	src/geom/remote_elem.C src/geom/sphere.C src/geom/surface.C \
@@ -715,6 +716,7 @@ am__objects_1 = src/base/libmesh_dbg_la-default_coupling.lo \
 	src/geom/libmesh_dbg_la-edge_edge4.lo \
 	src/geom/libmesh_dbg_la-edge_inf_edge2.lo \
 	src/geom/libmesh_dbg_la-elem.lo \
+	src/geom/libmesh_dbg_la-elem_corner.lo \
 	src/geom/libmesh_dbg_la-elem_cutter.lo \
 	src/geom/libmesh_dbg_la-elem_quality.lo \
 	src/geom/libmesh_dbg_la-elem_refinement.lo \
@@ -735,6 +737,7 @@ am__objects_1 = src/base/libmesh_dbg_la-default_coupling.lo \
 	src/geom/libmesh_dbg_la-face_tri3_subdivision.lo \
 	src/geom/libmesh_dbg_la-face_tri6.lo \
 	src/geom/libmesh_dbg_la-face_tri7.lo \
+	src/geom/libmesh_dbg_la-intersection_tools.lo \
 	src/geom/libmesh_dbg_la-node.lo \
 	src/geom/libmesh_dbg_la-node_elem.lo \
 	src/geom/libmesh_dbg_la-plane.lo \
@@ -1107,16 +1110,17 @@ am__libmesh_devel_la_SOURCES_DIST = src/base/default_coupling.C \
 	src/geom/cell_tet4.C src/geom/edge.C src/geom/edge_edge2.C \
 	src/geom/edge_edge3.C src/geom/edge_edge4.C \
 	src/geom/edge_inf_edge2.C src/geom/elem.C \
-	src/geom/elem_cutter.C src/geom/elem_quality.C \
-	src/geom/elem_refinement.C src/geom/elem_side_builder.C \
-	src/geom/face.C src/geom/face_inf_quad.C \
-	src/geom/face_inf_quad4.C src/geom/face_inf_quad6.C \
-	src/geom/face_quad.C src/geom/face_quad4.C \
-	src/geom/face_quad4_shell.C src/geom/face_quad8.C \
-	src/geom/face_quad8_shell.C src/geom/face_quad9.C \
-	src/geom/face_tri.C src/geom/face_tri3.C \
+	src/geom/elem_corner.C src/geom/elem_cutter.C \
+	src/geom/elem_quality.C src/geom/elem_refinement.C \
+	src/geom/elem_side_builder.C src/geom/face.C \
+	src/geom/face_inf_quad.C src/geom/face_inf_quad4.C \
+	src/geom/face_inf_quad6.C src/geom/face_quad.C \
+	src/geom/face_quad4.C src/geom/face_quad4_shell.C \
+	src/geom/face_quad8.C src/geom/face_quad8_shell.C \
+	src/geom/face_quad9.C src/geom/face_tri.C src/geom/face_tri3.C \
 	src/geom/face_tri3_shell.C src/geom/face_tri3_subdivision.C \
-	src/geom/face_tri6.C src/geom/face_tri7.C src/geom/node.C \
+	src/geom/face_tri6.C src/geom/face_tri7.C \
+	src/geom/intersection_tools.C src/geom/node.C \
 	src/geom/node_elem.C src/geom/plane.C src/geom/point.C \
 	src/geom/reference_elem.C src/geom/reference_elem.data \
 	src/geom/remote_elem.C src/geom/sphere.C src/geom/surface.C \
@@ -1475,6 +1479,7 @@ am__objects_2 = src/base/libmesh_devel_la-default_coupling.lo \
 	src/geom/libmesh_devel_la-edge_edge4.lo \
 	src/geom/libmesh_devel_la-edge_inf_edge2.lo \
 	src/geom/libmesh_devel_la-elem.lo \
+	src/geom/libmesh_devel_la-elem_corner.lo \
 	src/geom/libmesh_devel_la-elem_cutter.lo \
 	src/geom/libmesh_devel_la-elem_quality.lo \
 	src/geom/libmesh_devel_la-elem_refinement.lo \
@@ -1495,6 +1500,7 @@ am__objects_2 = src/base/libmesh_devel_la-default_coupling.lo \
 	src/geom/libmesh_devel_la-face_tri3_subdivision.lo \
 	src/geom/libmesh_devel_la-face_tri6.lo \
 	src/geom/libmesh_devel_la-face_tri7.lo \
+	src/geom/libmesh_devel_la-intersection_tools.lo \
 	src/geom/libmesh_devel_la-node.lo \
 	src/geom/libmesh_devel_la-node_elem.lo \
 	src/geom/libmesh_devel_la-plane.lo \
@@ -1864,16 +1870,17 @@ am__libmesh_oprof_la_SOURCES_DIST = src/base/default_coupling.C \
 	src/geom/cell_tet4.C src/geom/edge.C src/geom/edge_edge2.C \
 	src/geom/edge_edge3.C src/geom/edge_edge4.C \
 	src/geom/edge_inf_edge2.C src/geom/elem.C \
-	src/geom/elem_cutter.C src/geom/elem_quality.C \
-	src/geom/elem_refinement.C src/geom/elem_side_builder.C \
-	src/geom/face.C src/geom/face_inf_quad.C \
-	src/geom/face_inf_quad4.C src/geom/face_inf_quad6.C \
-	src/geom/face_quad.C src/geom/face_quad4.C \
-	src/geom/face_quad4_shell.C src/geom/face_quad8.C \
-	src/geom/face_quad8_shell.C src/geom/face_quad9.C \
-	src/geom/face_tri.C src/geom/face_tri3.C \
+	src/geom/elem_corner.C src/geom/elem_cutter.C \
+	src/geom/elem_quality.C src/geom/elem_refinement.C \
+	src/geom/elem_side_builder.C src/geom/face.C \
+	src/geom/face_inf_quad.C src/geom/face_inf_quad4.C \
+	src/geom/face_inf_quad6.C src/geom/face_quad.C \
+	src/geom/face_quad4.C src/geom/face_quad4_shell.C \
+	src/geom/face_quad8.C src/geom/face_quad8_shell.C \
+	src/geom/face_quad9.C src/geom/face_tri.C src/geom/face_tri3.C \
 	src/geom/face_tri3_shell.C src/geom/face_tri3_subdivision.C \
-	src/geom/face_tri6.C src/geom/face_tri7.C src/geom/node.C \
+	src/geom/face_tri6.C src/geom/face_tri7.C \
+	src/geom/intersection_tools.C src/geom/node.C \
 	src/geom/node_elem.C src/geom/plane.C src/geom/point.C \
 	src/geom/reference_elem.C src/geom/reference_elem.data \
 	src/geom/remote_elem.C src/geom/sphere.C src/geom/surface.C \
@@ -2232,6 +2239,7 @@ am__objects_3 = src/base/libmesh_oprof_la-default_coupling.lo \
 	src/geom/libmesh_oprof_la-edge_edge4.lo \
 	src/geom/libmesh_oprof_la-edge_inf_edge2.lo \
 	src/geom/libmesh_oprof_la-elem.lo \
+	src/geom/libmesh_oprof_la-elem_corner.lo \
 	src/geom/libmesh_oprof_la-elem_cutter.lo \
 	src/geom/libmesh_oprof_la-elem_quality.lo \
 	src/geom/libmesh_oprof_la-elem_refinement.lo \
@@ -2252,6 +2260,7 @@ am__objects_3 = src/base/libmesh_oprof_la-default_coupling.lo \
 	src/geom/libmesh_oprof_la-face_tri3_subdivision.lo \
 	src/geom/libmesh_oprof_la-face_tri6.lo \
 	src/geom/libmesh_oprof_la-face_tri7.lo \
+	src/geom/libmesh_oprof_la-intersection_tools.lo \
 	src/geom/libmesh_oprof_la-node.lo \
 	src/geom/libmesh_oprof_la-node_elem.lo \
 	src/geom/libmesh_oprof_la-plane.lo \
@@ -2621,16 +2630,17 @@ am__libmesh_opt_la_SOURCES_DIST = src/base/default_coupling.C \
 	src/geom/cell_tet4.C src/geom/edge.C src/geom/edge_edge2.C \
 	src/geom/edge_edge3.C src/geom/edge_edge4.C \
 	src/geom/edge_inf_edge2.C src/geom/elem.C \
-	src/geom/elem_cutter.C src/geom/elem_quality.C \
-	src/geom/elem_refinement.C src/geom/elem_side_builder.C \
-	src/geom/face.C src/geom/face_inf_quad.C \
-	src/geom/face_inf_quad4.C src/geom/face_inf_quad6.C \
-	src/geom/face_quad.C src/geom/face_quad4.C \
-	src/geom/face_quad4_shell.C src/geom/face_quad8.C \
-	src/geom/face_quad8_shell.C src/geom/face_quad9.C \
-	src/geom/face_tri.C src/geom/face_tri3.C \
+	src/geom/elem_corner.C src/geom/elem_cutter.C \
+	src/geom/elem_quality.C src/geom/elem_refinement.C \
+	src/geom/elem_side_builder.C src/geom/face.C \
+	src/geom/face_inf_quad.C src/geom/face_inf_quad4.C \
+	src/geom/face_inf_quad6.C src/geom/face_quad.C \
+	src/geom/face_quad4.C src/geom/face_quad4_shell.C \
+	src/geom/face_quad8.C src/geom/face_quad8_shell.C \
+	src/geom/face_quad9.C src/geom/face_tri.C src/geom/face_tri3.C \
 	src/geom/face_tri3_shell.C src/geom/face_tri3_subdivision.C \
-	src/geom/face_tri6.C src/geom/face_tri7.C src/geom/node.C \
+	src/geom/face_tri6.C src/geom/face_tri7.C \
+	src/geom/intersection_tools.C src/geom/node.C \
 	src/geom/node_elem.C src/geom/plane.C src/geom/point.C \
 	src/geom/reference_elem.C src/geom/reference_elem.data \
 	src/geom/remote_elem.C src/geom/sphere.C src/geom/surface.C \
@@ -2989,6 +2999,7 @@ am__objects_4 = src/base/libmesh_opt_la-default_coupling.lo \
 	src/geom/libmesh_opt_la-edge_edge4.lo \
 	src/geom/libmesh_opt_la-edge_inf_edge2.lo \
 	src/geom/libmesh_opt_la-elem.lo \
+	src/geom/libmesh_opt_la-elem_corner.lo \
 	src/geom/libmesh_opt_la-elem_cutter.lo \
 	src/geom/libmesh_opt_la-elem_quality.lo \
 	src/geom/libmesh_opt_la-elem_refinement.lo \
@@ -3009,6 +3020,7 @@ am__objects_4 = src/base/libmesh_opt_la-default_coupling.lo \
 	src/geom/libmesh_opt_la-face_tri3_subdivision.lo \
 	src/geom/libmesh_opt_la-face_tri6.lo \
 	src/geom/libmesh_opt_la-face_tri7.lo \
+	src/geom/libmesh_opt_la-intersection_tools.lo \
 	src/geom/libmesh_opt_la-node.lo \
 	src/geom/libmesh_opt_la-node_elem.lo \
 	src/geom/libmesh_opt_la-plane.lo \
@@ -3377,16 +3389,17 @@ am__libmesh_prof_la_SOURCES_DIST = src/base/default_coupling.C \
 	src/geom/cell_tet4.C src/geom/edge.C src/geom/edge_edge2.C \
 	src/geom/edge_edge3.C src/geom/edge_edge4.C \
 	src/geom/edge_inf_edge2.C src/geom/elem.C \
-	src/geom/elem_cutter.C src/geom/elem_quality.C \
-	src/geom/elem_refinement.C src/geom/elem_side_builder.C \
-	src/geom/face.C src/geom/face_inf_quad.C \
-	src/geom/face_inf_quad4.C src/geom/face_inf_quad6.C \
-	src/geom/face_quad.C src/geom/face_quad4.C \
-	src/geom/face_quad4_shell.C src/geom/face_quad8.C \
-	src/geom/face_quad8_shell.C src/geom/face_quad9.C \
-	src/geom/face_tri.C src/geom/face_tri3.C \
+	src/geom/elem_corner.C src/geom/elem_cutter.C \
+	src/geom/elem_quality.C src/geom/elem_refinement.C \
+	src/geom/elem_side_builder.C src/geom/face.C \
+	src/geom/face_inf_quad.C src/geom/face_inf_quad4.C \
+	src/geom/face_inf_quad6.C src/geom/face_quad.C \
+	src/geom/face_quad4.C src/geom/face_quad4_shell.C \
+	src/geom/face_quad8.C src/geom/face_quad8_shell.C \
+	src/geom/face_quad9.C src/geom/face_tri.C src/geom/face_tri3.C \
 	src/geom/face_tri3_shell.C src/geom/face_tri3_subdivision.C \
-	src/geom/face_tri6.C src/geom/face_tri7.C src/geom/node.C \
+	src/geom/face_tri6.C src/geom/face_tri7.C \
+	src/geom/intersection_tools.C src/geom/node.C \
 	src/geom/node_elem.C src/geom/plane.C src/geom/point.C \
 	src/geom/reference_elem.C src/geom/reference_elem.data \
 	src/geom/remote_elem.C src/geom/sphere.C src/geom/surface.C \
@@ -3745,6 +3758,7 @@ am__objects_5 = src/base/libmesh_prof_la-default_coupling.lo \
 	src/geom/libmesh_prof_la-edge_edge4.lo \
 	src/geom/libmesh_prof_la-edge_inf_edge2.lo \
 	src/geom/libmesh_prof_la-elem.lo \
+	src/geom/libmesh_prof_la-elem_corner.lo \
 	src/geom/libmesh_prof_la-elem_cutter.lo \
 	src/geom/libmesh_prof_la-elem_quality.lo \
 	src/geom/libmesh_prof_la-elem_refinement.lo \
@@ -3765,6 +3779,7 @@ am__objects_5 = src/base/libmesh_prof_la-default_coupling.lo \
 	src/geom/libmesh_prof_la-face_tri3_subdivision.lo \
 	src/geom/libmesh_prof_la-face_tri6.lo \
 	src/geom/libmesh_prof_la-face_tri7.lo \
+	src/geom/libmesh_prof_la-intersection_tools.lo \
 	src/geom/libmesh_prof_la-node.lo \
 	src/geom/libmesh_prof_la-node_elem.lo \
 	src/geom/libmesh_prof_la-plane.lo \
@@ -5104,6 +5119,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-edge_edge4.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-edge_inf_edge2.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-elem.Plo \
+	src/geom/$(DEPDIR)/libmesh_dbg_la-elem_corner.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-elem_cutter.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-elem_quality.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-elem_refinement.Plo \
@@ -5124,6 +5140,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri3_subdivision.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri6.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri7.Plo \
+	src/geom/$(DEPDIR)/libmesh_dbg_la-intersection_tools.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-node.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-node_elem.Plo \
 	src/geom/$(DEPDIR)/libmesh_dbg_la-plane.Plo \
@@ -5166,6 +5183,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_devel_la-edge_edge4.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-edge_inf_edge2.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-elem.Plo \
+	src/geom/$(DEPDIR)/libmesh_devel_la-elem_corner.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-elem_cutter.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-elem_quality.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-elem_refinement.Plo \
@@ -5186,6 +5204,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_devel_la-face_tri3_subdivision.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-face_tri6.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-face_tri7.Plo \
+	src/geom/$(DEPDIR)/libmesh_devel_la-intersection_tools.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-node.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-node_elem.Plo \
 	src/geom/$(DEPDIR)/libmesh_devel_la-plane.Plo \
@@ -5228,6 +5247,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-edge_edge4.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-edge_inf_edge2.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-elem.Plo \
+	src/geom/$(DEPDIR)/libmesh_oprof_la-elem_corner.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-elem_cutter.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-elem_quality.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-elem_refinement.Plo \
@@ -5248,6 +5268,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri3_subdivision.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri6.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri7.Plo \
+	src/geom/$(DEPDIR)/libmesh_oprof_la-intersection_tools.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-node.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-node_elem.Plo \
 	src/geom/$(DEPDIR)/libmesh_oprof_la-plane.Plo \
@@ -5290,6 +5311,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_opt_la-edge_edge4.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-edge_inf_edge2.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-elem.Plo \
+	src/geom/$(DEPDIR)/libmesh_opt_la-elem_corner.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-elem_cutter.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-elem_quality.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-elem_refinement.Plo \
@@ -5310,6 +5332,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_opt_la-face_tri3_subdivision.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-face_tri6.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-face_tri7.Plo \
+	src/geom/$(DEPDIR)/libmesh_opt_la-intersection_tools.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-node.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-node_elem.Plo \
 	src/geom/$(DEPDIR)/libmesh_opt_la-plane.Plo \
@@ -5352,6 +5375,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_prof_la-edge_edge4.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-edge_inf_edge2.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-elem.Plo \
+	src/geom/$(DEPDIR)/libmesh_prof_la-elem_corner.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-elem_cutter.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-elem_quality.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-elem_refinement.Plo \
@@ -5372,6 +5396,7 @@ am__depfiles_remade = src/apps/$(DEPDIR)/amr_dbg-amr.Po \
 	src/geom/$(DEPDIR)/libmesh_prof_la-face_tri3_subdivision.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-face_tri6.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-face_tri7.Plo \
+	src/geom/$(DEPDIR)/libmesh_prof_la-intersection_tools.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-node.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-node_elem.Plo \
 	src/geom/$(DEPDIR)/libmesh_prof_la-plane.Plo \
@@ -7508,6 +7533,7 @@ libmesh_SOURCES = \
         src/geom/edge_edge4.C \
         src/geom/edge_inf_edge2.C \
         src/geom/elem.C \
+        src/geom/elem_corner.C \
         src/geom/elem_cutter.C \
         src/geom/elem_quality.C \
         src/geom/elem_refinement.C \
@@ -7528,6 +7554,7 @@ libmesh_SOURCES = \
         src/geom/face_tri3_subdivision.C \
         src/geom/face_tri6.C \
         src/geom/face_tri7.C \
+        src/geom/intersection_tools.C \
         src/geom/node.C \
         src/geom/node_elem.C \
         src/geom/plane.C \
@@ -8589,6 +8616,8 @@ src/geom/libmesh_dbg_la-edge_inf_edge2.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_dbg_la-elem.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_dbg_la-elem_corner.lo: src/geom/$(am__dirstamp) \
+	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_dbg_la-elem_cutter.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_dbg_la-elem_quality.lo: src/geom/$(am__dirstamp) \
@@ -8629,6 +8658,8 @@ src/geom/libmesh_dbg_la-face_tri6.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_dbg_la-face_tri7.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_dbg_la-intersection_tools.lo:  \
+	src/geom/$(am__dirstamp) src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_dbg_la-node.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_dbg_la-node_elem.lo: src/geom/$(am__dirstamp) \
@@ -9752,6 +9783,8 @@ src/geom/libmesh_devel_la-edge_inf_edge2.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_devel_la-elem.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_devel_la-elem_corner.lo: src/geom/$(am__dirstamp) \
+	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_devel_la-elem_cutter.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_devel_la-elem_quality.lo: src/geom/$(am__dirstamp) \
@@ -9792,6 +9825,8 @@ src/geom/libmesh_devel_la-face_tri6.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_devel_la-face_tri7.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_devel_la-intersection_tools.lo:  \
+	src/geom/$(am__dirstamp) src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_devel_la-node.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_devel_la-node_elem.lo: src/geom/$(am__dirstamp) \
@@ -10852,6 +10887,8 @@ src/geom/libmesh_oprof_la-edge_inf_edge2.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_oprof_la-elem.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_oprof_la-elem_corner.lo: src/geom/$(am__dirstamp) \
+	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_oprof_la-elem_cutter.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_oprof_la-elem_quality.lo: src/geom/$(am__dirstamp) \
@@ -10892,6 +10929,8 @@ src/geom/libmesh_oprof_la-face_tri6.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_oprof_la-face_tri7.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_oprof_la-intersection_tools.lo:  \
+	src/geom/$(am__dirstamp) src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_oprof_la-node.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_oprof_la-node_elem.lo: src/geom/$(am__dirstamp) \
@@ -11952,6 +11991,8 @@ src/geom/libmesh_opt_la-edge_inf_edge2.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_opt_la-elem.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_opt_la-elem_corner.lo: src/geom/$(am__dirstamp) \
+	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_opt_la-elem_cutter.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_opt_la-elem_quality.lo: src/geom/$(am__dirstamp) \
@@ -11992,6 +12033,8 @@ src/geom/libmesh_opt_la-face_tri6.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_opt_la-face_tri7.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_opt_la-intersection_tools.lo:  \
+	src/geom/$(am__dirstamp) src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_opt_la-node.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_opt_la-node_elem.lo: src/geom/$(am__dirstamp) \
@@ -13049,6 +13092,8 @@ src/geom/libmesh_prof_la-edge_inf_edge2.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_prof_la-elem.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_prof_la-elem_corner.lo: src/geom/$(am__dirstamp) \
+	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_prof_la-elem_cutter.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_prof_la-elem_quality.lo: src/geom/$(am__dirstamp) \
@@ -13089,6 +13134,8 @@ src/geom/libmesh_prof_la-face_tri6.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_prof_la-face_tri7.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
+src/geom/libmesh_prof_la-intersection_tools.lo:  \
+	src/geom/$(am__dirstamp) src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_prof_la-node.lo: src/geom/$(am__dirstamp) \
 	src/geom/$(DEPDIR)/$(am__dirstamp)
 src/geom/libmesh_prof_la-node_elem.lo: src/geom/$(am__dirstamp) \
@@ -14967,6 +15014,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-edge_edge4.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-edge_inf_edge2.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-elem.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-elem_corner.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-elem_cutter.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-elem_quality.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-elem_refinement.Plo@am__quote@ # am--include-marker
@@ -14987,6 +15035,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri3_subdivision.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri6.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri7.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-intersection_tools.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-node.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-node_elem.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_dbg_la-plane.Plo@am__quote@ # am--include-marker
@@ -15029,6 +15078,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-edge_edge4.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-edge_inf_edge2.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-elem.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-elem_corner.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-elem_cutter.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-elem_quality.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-elem_refinement.Plo@am__quote@ # am--include-marker
@@ -15049,6 +15099,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-face_tri3_subdivision.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-face_tri6.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-face_tri7.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-intersection_tools.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-node.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-node_elem.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_devel_la-plane.Plo@am__quote@ # am--include-marker
@@ -15091,6 +15142,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-edge_edge4.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-edge_inf_edge2.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-elem.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-elem_corner.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-elem_cutter.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-elem_quality.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-elem_refinement.Plo@am__quote@ # am--include-marker
@@ -15111,6 +15163,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri3_subdivision.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri6.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri7.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-intersection_tools.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-node.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-node_elem.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_oprof_la-plane.Plo@am__quote@ # am--include-marker
@@ -15153,6 +15206,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-edge_edge4.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-edge_inf_edge2.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-elem.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-elem_corner.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-elem_cutter.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-elem_quality.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-elem_refinement.Plo@am__quote@ # am--include-marker
@@ -15173,6 +15227,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-face_tri3_subdivision.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-face_tri6.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-face_tri7.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-intersection_tools.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-node.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-node_elem.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_opt_la-plane.Plo@am__quote@ # am--include-marker
@@ -15215,6 +15270,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-edge_edge4.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-edge_inf_edge2.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-elem.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-elem_corner.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-elem_cutter.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-elem_quality.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-elem_refinement.Plo@am__quote@ # am--include-marker
@@ -15235,6 +15291,7 @@ distclean-compile:
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-face_tri3_subdivision.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-face_tri6.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-face_tri7.Plo@am__quote@ # am--include-marker
+@AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-intersection_tools.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-node.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-node_elem.Plo@am__quote@ # am--include-marker
 @AMDEP_TRUE@@am__include@ @am__quote@src/geom/$(DEPDIR)/libmesh_prof_la-plane.Plo@am__quote@ # am--include-marker
@@ -17668,6 +17725,13 @@ src/geom/libmesh_dbg_la-elem.lo: src/geom/elem.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_dbg_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_dbg_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_dbg_la-elem.lo `test -f 'src/geom/elem.C' || echo '$(srcdir)/'`src/geom/elem.C
 
+src/geom/libmesh_dbg_la-elem_corner.lo: src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_dbg_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_dbg_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_dbg_la-elem_corner.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_dbg_la-elem_corner.Tpo -c -o src/geom/libmesh_dbg_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_dbg_la-elem_corner.Tpo src/geom/$(DEPDIR)/libmesh_dbg_la-elem_corner.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/elem_corner.C' object='src/geom/libmesh_dbg_la-elem_corner.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_dbg_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_dbg_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_dbg_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+
 src/geom/libmesh_dbg_la-elem_cutter.lo: src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_dbg_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_dbg_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_dbg_la-elem_cutter.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_dbg_la-elem_cutter.Tpo -c -o src/geom/libmesh_dbg_la-elem_cutter.lo `test -f 'src/geom/elem_cutter.C' || echo '$(srcdir)/'`src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_dbg_la-elem_cutter.Tpo src/geom/$(DEPDIR)/libmesh_dbg_la-elem_cutter.Plo
@@ -17808,6 +17872,13 @@ src/geom/libmesh_dbg_la-face_tri7.lo: src/geom/face_tri7.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_dbg_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_dbg_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_dbg_la-face_tri7.lo `test -f 'src/geom/face_tri7.C' || echo '$(srcdir)/'`src/geom/face_tri7.C
 
+src/geom/libmesh_dbg_la-intersection_tools.lo: src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_dbg_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_dbg_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_dbg_la-intersection_tools.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_dbg_la-intersection_tools.Tpo -c -o src/geom/libmesh_dbg_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_dbg_la-intersection_tools.Tpo src/geom/$(DEPDIR)/libmesh_dbg_la-intersection_tools.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/intersection_tools.C' object='src/geom/libmesh_dbg_la-intersection_tools.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_dbg_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_dbg_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_dbg_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+
 src/geom/libmesh_dbg_la-node.lo: src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_dbg_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_dbg_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_dbg_la-node.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_dbg_la-node.Tpo -c -o src/geom/libmesh_dbg_la-node.lo `test -f 'src/geom/node.C' || echo '$(srcdir)/'`src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_dbg_la-node.Tpo src/geom/$(DEPDIR)/libmesh_dbg_la-node.Plo
@@ -20790,6 +20861,13 @@ src/geom/libmesh_devel_la-elem.lo: src/geom/elem.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_devel_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_devel_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_devel_la-elem.lo `test -f 'src/geom/elem.C' || echo '$(srcdir)/'`src/geom/elem.C
 
+src/geom/libmesh_devel_la-elem_corner.lo: src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_devel_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_devel_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_devel_la-elem_corner.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_devel_la-elem_corner.Tpo -c -o src/geom/libmesh_devel_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_devel_la-elem_corner.Tpo src/geom/$(DEPDIR)/libmesh_devel_la-elem_corner.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/elem_corner.C' object='src/geom/libmesh_devel_la-elem_corner.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_devel_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_devel_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_devel_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+
 src/geom/libmesh_devel_la-elem_cutter.lo: src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_devel_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_devel_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_devel_la-elem_cutter.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_devel_la-elem_cutter.Tpo -c -o src/geom/libmesh_devel_la-elem_cutter.lo `test -f 'src/geom/elem_cutter.C' || echo '$(srcdir)/'`src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_devel_la-elem_cutter.Tpo src/geom/$(DEPDIR)/libmesh_devel_la-elem_cutter.Plo
@@ -20930,6 +21008,13 @@ src/geom/libmesh_devel_la-face_tri7.lo: src/geom/face_tri7.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_devel_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_devel_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_devel_la-face_tri7.lo `test -f 'src/geom/face_tri7.C' || echo '$(srcdir)/'`src/geom/face_tri7.C
 
+src/geom/libmesh_devel_la-intersection_tools.lo: src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_devel_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_devel_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_devel_la-intersection_tools.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_devel_la-intersection_tools.Tpo -c -o src/geom/libmesh_devel_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_devel_la-intersection_tools.Tpo src/geom/$(DEPDIR)/libmesh_devel_la-intersection_tools.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/intersection_tools.C' object='src/geom/libmesh_devel_la-intersection_tools.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_devel_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_devel_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_devel_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+
 src/geom/libmesh_devel_la-node.lo: src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_devel_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_devel_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_devel_la-node.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_devel_la-node.Tpo -c -o src/geom/libmesh_devel_la-node.lo `test -f 'src/geom/node.C' || echo '$(srcdir)/'`src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_devel_la-node.Tpo src/geom/$(DEPDIR)/libmesh_devel_la-node.Plo
@@ -23912,6 +23997,13 @@ src/geom/libmesh_oprof_la-elem.lo: src/geom/elem.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_oprof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_oprof_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_oprof_la-elem.lo `test -f 'src/geom/elem.C' || echo '$(srcdir)/'`src/geom/elem.C
 
+src/geom/libmesh_oprof_la-elem_corner.lo: src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_oprof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_oprof_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_oprof_la-elem_corner.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_oprof_la-elem_corner.Tpo -c -o src/geom/libmesh_oprof_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_oprof_la-elem_corner.Tpo src/geom/$(DEPDIR)/libmesh_oprof_la-elem_corner.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/elem_corner.C' object='src/geom/libmesh_oprof_la-elem_corner.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_oprof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_oprof_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_oprof_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+
 src/geom/libmesh_oprof_la-elem_cutter.lo: src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_oprof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_oprof_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_oprof_la-elem_cutter.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_oprof_la-elem_cutter.Tpo -c -o src/geom/libmesh_oprof_la-elem_cutter.lo `test -f 'src/geom/elem_cutter.C' || echo '$(srcdir)/'`src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_oprof_la-elem_cutter.Tpo src/geom/$(DEPDIR)/libmesh_oprof_la-elem_cutter.Plo
@@ -24052,6 +24144,13 @@ src/geom/libmesh_oprof_la-face_tri7.lo: src/geom/face_tri7.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_oprof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_oprof_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_oprof_la-face_tri7.lo `test -f 'src/geom/face_tri7.C' || echo '$(srcdir)/'`src/geom/face_tri7.C
 
+src/geom/libmesh_oprof_la-intersection_tools.lo: src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_oprof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_oprof_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_oprof_la-intersection_tools.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_oprof_la-intersection_tools.Tpo -c -o src/geom/libmesh_oprof_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_oprof_la-intersection_tools.Tpo src/geom/$(DEPDIR)/libmesh_oprof_la-intersection_tools.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/intersection_tools.C' object='src/geom/libmesh_oprof_la-intersection_tools.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_oprof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_oprof_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_oprof_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+
 src/geom/libmesh_oprof_la-node.lo: src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_oprof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_oprof_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_oprof_la-node.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_oprof_la-node.Tpo -c -o src/geom/libmesh_oprof_la-node.lo `test -f 'src/geom/node.C' || echo '$(srcdir)/'`src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_oprof_la-node.Tpo src/geom/$(DEPDIR)/libmesh_oprof_la-node.Plo
@@ -27034,6 +27133,13 @@ src/geom/libmesh_opt_la-elem.lo: src/geom/elem.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_opt_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_opt_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_opt_la-elem.lo `test -f 'src/geom/elem.C' || echo '$(srcdir)/'`src/geom/elem.C
 
+src/geom/libmesh_opt_la-elem_corner.lo: src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_opt_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_opt_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_opt_la-elem_corner.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_opt_la-elem_corner.Tpo -c -o src/geom/libmesh_opt_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_opt_la-elem_corner.Tpo src/geom/$(DEPDIR)/libmesh_opt_la-elem_corner.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/elem_corner.C' object='src/geom/libmesh_opt_la-elem_corner.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_opt_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_opt_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_opt_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+
 src/geom/libmesh_opt_la-elem_cutter.lo: src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_opt_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_opt_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_opt_la-elem_cutter.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_opt_la-elem_cutter.Tpo -c -o src/geom/libmesh_opt_la-elem_cutter.lo `test -f 'src/geom/elem_cutter.C' || echo '$(srcdir)/'`src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_opt_la-elem_cutter.Tpo src/geom/$(DEPDIR)/libmesh_opt_la-elem_cutter.Plo
@@ -27174,6 +27280,13 @@ src/geom/libmesh_opt_la-face_tri7.lo: src/geom/face_tri7.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_opt_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_opt_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_opt_la-face_tri7.lo `test -f 'src/geom/face_tri7.C' || echo '$(srcdir)/'`src/geom/face_tri7.C
 
+src/geom/libmesh_opt_la-intersection_tools.lo: src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_opt_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_opt_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_opt_la-intersection_tools.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_opt_la-intersection_tools.Tpo -c -o src/geom/libmesh_opt_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_opt_la-intersection_tools.Tpo src/geom/$(DEPDIR)/libmesh_opt_la-intersection_tools.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/intersection_tools.C' object='src/geom/libmesh_opt_la-intersection_tools.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_opt_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_opt_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_opt_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+
 src/geom/libmesh_opt_la-node.lo: src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_opt_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_opt_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_opt_la-node.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_opt_la-node.Tpo -c -o src/geom/libmesh_opt_la-node.lo `test -f 'src/geom/node.C' || echo '$(srcdir)/'`src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_opt_la-node.Tpo src/geom/$(DEPDIR)/libmesh_opt_la-node.Plo
@@ -30156,6 +30269,13 @@ src/geom/libmesh_prof_la-elem.lo: src/geom/elem.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_prof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_prof_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_prof_la-elem.lo `test -f 'src/geom/elem.C' || echo '$(srcdir)/'`src/geom/elem.C
 
+src/geom/libmesh_prof_la-elem_corner.lo: src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_prof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_prof_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_prof_la-elem_corner.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_prof_la-elem_corner.Tpo -c -o src/geom/libmesh_prof_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_prof_la-elem_corner.Tpo src/geom/$(DEPDIR)/libmesh_prof_la-elem_corner.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/elem_corner.C' object='src/geom/libmesh_prof_la-elem_corner.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_prof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_prof_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_prof_la-elem_corner.lo `test -f 'src/geom/elem_corner.C' || echo '$(srcdir)/'`src/geom/elem_corner.C
+
 src/geom/libmesh_prof_la-elem_cutter.lo: src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_prof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_prof_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_prof_la-elem_cutter.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_prof_la-elem_cutter.Tpo -c -o src/geom/libmesh_prof_la-elem_cutter.lo `test -f 'src/geom/elem_cutter.C' || echo '$(srcdir)/'`src/geom/elem_cutter.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_prof_la-elem_cutter.Tpo src/geom/$(DEPDIR)/libmesh_prof_la-elem_cutter.Plo
@@ -30296,6 +30416,13 @@ src/geom/libmesh_prof_la-face_tri7.lo: src/geom/face_tri7.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_prof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_prof_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_prof_la-face_tri7.lo `test -f 'src/geom/face_tri7.C' || echo '$(srcdir)/'`src/geom/face_tri7.C
 
+src/geom/libmesh_prof_la-intersection_tools.lo: src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_prof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_prof_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_prof_la-intersection_tools.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_prof_la-intersection_tools.Tpo -c -o src/geom/libmesh_prof_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_prof_la-intersection_tools.Tpo src/geom/$(DEPDIR)/libmesh_prof_la-intersection_tools.Plo
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='src/geom/intersection_tools.C' object='src/geom/libmesh_prof_la-intersection_tools.lo' libtool=yes @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_prof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_prof_la_CXXFLAGS) $(CXXFLAGS) -c -o src/geom/libmesh_prof_la-intersection_tools.lo `test -f 'src/geom/intersection_tools.C' || echo '$(srcdir)/'`src/geom/intersection_tools.C
+
 src/geom/libmesh_prof_la-node.lo: src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(LIBTOOL) $(AM_V_lt) --tag=CXX $(AM_LIBTOOLFLAGS) $(LIBTOOLFLAGS) --mode=compile $(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(libmesh_prof_la_CPPFLAGS) $(CPPFLAGS) $(libmesh_prof_la_CXXFLAGS) $(CXXFLAGS) -MT src/geom/libmesh_prof_la-node.lo -MD -MP -MF src/geom/$(DEPDIR)/libmesh_prof_la-node.Tpo -c -o src/geom/libmesh_prof_la-node.lo `test -f 'src/geom/node.C' || echo '$(srcdir)/'`src/geom/node.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) src/geom/$(DEPDIR)/libmesh_prof_la-node.Tpo src/geom/$(DEPDIR)/libmesh_prof_la-node.Plo
@@ -34117,6 +34244,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem_refinement.Plo
@@ -34137,6 +34265,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-plane.Plo
@@ -34179,6 +34308,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem_refinement.Plo
@@ -34199,6 +34329,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-plane.Plo
@@ -34241,6 +34372,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem_refinement.Plo
@@ -34261,6 +34393,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-plane.Plo
@@ -34303,6 +34436,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem_refinement.Plo
@@ -34323,6 +34457,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-plane.Plo
@@ -34365,6 +34500,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem_refinement.Plo
@@ -34385,6 +34521,7 @@ distclean: distclean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-plane.Plo
@@ -36452,6 +36589,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-elem_refinement.Plo
@@ -36472,6 +36610,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_dbg_la-plane.Plo
@@ -36514,6 +36653,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-elem_refinement.Plo
@@ -36534,6 +36674,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_devel_la-plane.Plo
@@ -36576,6 +36717,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-elem_refinement.Plo
@@ -36596,6 +36738,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_oprof_la-plane.Plo
@@ -36638,6 +36781,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-elem_refinement.Plo
@@ -36658,6 +36802,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_opt_la-plane.Plo
@@ -36700,6 +36845,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-edge_edge4.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-edge_inf_edge2.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem_corner.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem_cutter.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem_quality.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-elem_refinement.Plo
@@ -36720,6 +36866,7 @@ maintainer-clean: maintainer-clean-recursive
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-face_tri3_subdivision.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-face_tri6.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-face_tri7.Plo
+	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-intersection_tools.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-node.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-node_elem.Plo
 	-rm -f src/geom/$(DEPDIR)/libmesh_prof_la-plane.Plo
diff --git a/include/Makefile.in b/include/Makefile.in
index 328648c0577..c31c7304227 100644
--- a/include/Makefile.in
+++ b/include/Makefile.in
@@ -729,6 +729,7 @@ include_HEADERS = \
         geom/edge_edge4.h \
         geom/edge_inf_edge2.h \
         geom/elem.h \
+        geom/elem_corner.h \
         geom/elem_cutter.h \
         geom/elem_hash.h \
         geom/elem_internal.h \
@@ -751,6 +752,7 @@ include_HEADERS = \
         geom/face_tri3_subdivision.h \
         geom/face_tri6.h \
         geom/face_tri7.h \
+        geom/intersection_tools.h \
         geom/node.h \
         geom/node_elem.h \
         geom/node_range.h \
diff --git a/include/geom/cell_hex.h b/include/geom/cell_hex.h
index 199ba4b1a36..423ea9de2a2 100644
--- a/include/geom/cell_hex.h
+++ b/include/geom/cell_hex.h
@@ -77,6 +77,12 @@ class Hex : public Cell
    */
   virtual unsigned int n_vertices() const override final { return 8; }
 
+  /**
+   * \returns 4. Every side has four vertices.
+   */
+  virtual unsigned int n_vertices_on_side(const unsigned short libmesh_dbg_var(s)) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 4; }
+
   /**
    * \returns 12.  All hexahedra have 12 edges.
    */
diff --git a/include/geom/cell_hex20.h b/include/geom/cell_hex20.h
index 18feb68a294..6fcb53f161f 100644
--- a/include/geom/cell_hex20.h
+++ b/include/geom/cell_hex20.h
@@ -124,8 +124,14 @@ class Hex20 final : public Hex
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Hex20>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Hex20>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -240,6 +246,18 @@ class Hex20 final : public Hex
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 8. Every side has eight nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Hex20>(s); }
+
+  /**
+   * \returns 3. Every edge has two nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Hex20>(e); }
+
 protected:
 
   /**
@@ -247,8 +265,6 @@ class Hex20 final : public Hex
    */
   Node * _nodelinks_data[num_nodes];
 
-
-
 #ifdef LIBMESH_ENABLE_AMR
 
   /**
diff --git a/include/geom/cell_hex27.h b/include/geom/cell_hex27.h
index 86bd5215d24..9f1d4c1568f 100644
--- a/include/geom/cell_hex27.h
+++ b/include/geom/cell_hex27.h
@@ -124,8 +124,14 @@ class Hex27 final : public Hex
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Hex27>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Hex27>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -258,6 +264,18 @@ class Hex27 final : public Hex
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 9. Every side has eight nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Hex27>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Hex27>(e); }
+
 protected:
 
 
diff --git a/include/geom/cell_hex8.h b/include/geom/cell_hex8.h
index 78e45f08858..09b5e643f8f 100644
--- a/include/geom/cell_hex8.h
+++ b/include/geom/cell_hex8.h
@@ -109,8 +109,14 @@ class Hex8 final : public Hex
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Hex8>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Hex8>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -215,6 +221,18 @@ class Hex8 final : public Hex
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 4. Every side has four nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Hex8>(s); }
+
+  /**
+   * \returns 2. Every edge has two nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Hex8>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_inf_hex.h b/include/geom/cell_inf_hex.h
index ce0dca8df52..3afaac58092 100644
--- a/include/geom/cell_inf_hex.h
+++ b/include/geom/cell_inf_hex.h
@@ -90,6 +90,12 @@ class InfHex : public InfCell
    */
   virtual unsigned int n_vertices() const override final { return 8; }
 
+  /**
+   * \returns 4. Every side has four vertices.
+   */
+  virtual unsigned int n_vertices_on_side(const unsigned short libmesh_dbg_var(s)) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 4; }
+
   /**
    * \returns \p true if the specified (local) node number is a
    * "mid-edge" node on an infinite element edge.
@@ -194,6 +200,19 @@ class InfHex : public InfCell
    */
   static const unsigned int edge_sides_map[8][2];
 
+  /**
+   * Helper for n_nodes_on_side() for InfHex-derived classes.
+   *
+   * That is, returns InfHex::nodes_per_side - \p remove_num
+   * for sides 1-4, and InfHex::nodes_per_side for side 0.
+  */
+  template <class InfHexClass, unsigned short remove_num>
+  unsigned int _n_nodes_on_side(const unsigned short s) const
+  {
+    static_assert(std::is_base_of<InfHex, InfHexClass>::value, "Not a InfHex");
+    return this->_n_nodes_on_side_constant<InfHexClass>(s) - ((s == 0) ? 0 : remove_num);
+  }
+
 protected:
 
   /**
diff --git a/include/geom/cell_inf_hex16.h b/include/geom/cell_inf_hex16.h
index c8419ba4602..36d4a302243 100644
--- a/include/geom/cell_inf_hex16.h
+++ b/include/geom/cell_inf_hex16.h
@@ -124,8 +124,14 @@ class InfHex16 final : public InfHex
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<InfHex16>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<InfHex16>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -238,6 +244,12 @@ class InfHex16 final : public InfHex
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side<InfHex16, 2>(s); }
+
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<InfHex16>(e) - ((e < 4) ? 0 : 1); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_inf_hex18.h b/include/geom/cell_inf_hex18.h
index d09dcc41d62..305fbb37193 100644
--- a/include/geom/cell_inf_hex18.h
+++ b/include/geom/cell_inf_hex18.h
@@ -129,8 +129,14 @@ class InfHex18 final : public InfHex
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<InfHex18>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<InfHex18>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -253,6 +259,12 @@ class InfHex18 final : public InfHex
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side<InfHex18, 3>(s); }
+
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<InfHex18>(e) - ((e < 4) ? 0 : 1); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_inf_hex8.h b/include/geom/cell_inf_hex8.h
index 590be2aba8b..89bafb6f47e 100644
--- a/include/geom/cell_inf_hex8.h
+++ b/include/geom/cell_inf_hex8.h
@@ -109,8 +109,14 @@ class InfHex8 final : public InfHex
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<InfHex8>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<InfHex8>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -188,6 +194,18 @@ class InfHex8 final : public InfHex
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 4. Every side has four nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<InfHex8>(s); }
+
+  /**
+   * \returns 2. Every edge has two nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<InfHex8>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_inf_prism.h b/include/geom/cell_inf_prism.h
index b80014014bc..338e91de3e8 100644
--- a/include/geom/cell_inf_prism.h
+++ b/include/geom/cell_inf_prism.h
@@ -86,6 +86,9 @@ class InfPrism : public InfCell
    */
   virtual unsigned int n_vertices() const override final { return 6; }
 
+  virtual unsigned int n_vertices_on_side(const unsigned short s) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 4 - (s == 0); }
+
   /**
    * \returns 6.  All infinite prisms have 6 edges,
    * 3 lying in the base, and 3 perpendicular to the base.
diff --git a/include/geom/cell_inf_prism12.h b/include/geom/cell_inf_prism12.h
index 6f9837ff52f..1c8c2a0b856 100644
--- a/include/geom/cell_inf_prism12.h
+++ b/include/geom/cell_inf_prism12.h
@@ -115,8 +115,14 @@ class InfPrism12 final : public InfPrism
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<InfPrism12>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<InfPrism12>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -226,6 +232,15 @@ class InfPrism12 final : public InfPrism
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 6. Every side has six nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<InfPrism12>(s); }
+
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<InfPrism12>(e) - ((e < 3) ? 0 : 1); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_inf_prism6.h b/include/geom/cell_inf_prism6.h
index 3b1100e6cc2..b9f6f5121ee 100644
--- a/include/geom/cell_inf_prism6.h
+++ b/include/geom/cell_inf_prism6.h
@@ -111,8 +111,14 @@ class InfPrism6 final : public InfPrism
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<InfPrism6>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<InfPrism6>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -187,6 +193,12 @@ class InfPrism6 final : public InfPrism
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<InfPrism6>(s) - ((s == 0) ? 1 : 0); }
+
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<InfPrism6>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_prism.h b/include/geom/cell_prism.h
index 06a23683302..735b740df26 100644
--- a/include/geom/cell_prism.h
+++ b/include/geom/cell_prism.h
@@ -83,6 +83,9 @@ class Prism : public Cell
    */
   virtual unsigned int n_vertices() const override final { return 6; }
 
+  virtual unsigned int n_vertices_on_side(const unsigned short s) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 4 - (s == 0 || s == 4); }
+
   /**
    * \returns 9.  All prisms have 9 edges.
    */
@@ -160,6 +163,19 @@ class Prism : public Cell
 
 protected:
 
+  /**
+   * Helper for n_nodes_on_side() for Prism-derived classes.
+   *
+   * That is, returns \p PrismClass::nodes_per_side for sides 1-3, and
+   * the same value minus \p remove_num for sides 0 and 4.
+  */
+  template <class PrismClass, unsigned short remove_num>
+  unsigned int _n_nodes_on_side(const unsigned short s) const
+  {
+    static_assert(std::is_base_of<Prism, PrismClass>::value, "Not a Prism");
+    return this->_n_nodes_on_side_constant<PrismClass>(s) - ((s == 0 || s == 4) ? remove_num : 0);
+  }
+
   /**
    * Data for links to parent/neighbor/interior_parent elements.
    */
diff --git a/include/geom/cell_prism15.h b/include/geom/cell_prism15.h
index 89c7ac56375..42b282105db 100644
--- a/include/geom/cell_prism15.h
+++ b/include/geom/cell_prism15.h
@@ -129,8 +129,14 @@ class Prism15 final : public Prism
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Prism15>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Prism15>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -245,6 +251,15 @@ class Prism15 final : public Prism
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return _n_nodes_on_side<Prism15, 2>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Prism15>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_prism18.h b/include/geom/cell_prism18.h
index fc646fa339c..7759d14a565 100644
--- a/include/geom/cell_prism18.h
+++ b/include/geom/cell_prism18.h
@@ -129,8 +129,14 @@ class Prism18 final : public Prism
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Prism18>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Prism18>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -263,6 +269,15 @@ class Prism18 final : public Prism
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return _n_nodes_on_side<Prism18, 3>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Prism18>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_prism20.h b/include/geom/cell_prism20.h
index e824e000c7f..021b2bdc33a 100644
--- a/include/geom/cell_prism20.h
+++ b/include/geom/cell_prism20.h
@@ -133,8 +133,14 @@ class Prism20 final : public Prism
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Prism20>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Prism20>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -262,6 +268,15 @@ class Prism20 final : public Prism
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return _n_nodes_on_side<Prism20, 2>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Prism20>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_prism21.h b/include/geom/cell_prism21.h
index c36da8469d1..eac0d9a187e 100644
--- a/include/geom/cell_prism21.h
+++ b/include/geom/cell_prism21.h
@@ -136,8 +136,14 @@ class Prism21 final : public Prism
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Prism21>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Prism21>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -265,6 +271,15 @@ class Prism21 final : public Prism
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return _n_nodes_on_side<Prism21, 2>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Prism21>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_prism6.h b/include/geom/cell_prism6.h
index 5009bffda9b..0213ef2c343 100644
--- a/include/geom/cell_prism6.h
+++ b/include/geom/cell_prism6.h
@@ -107,8 +107,14 @@ class Prism6 final : public Prism
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Prism6>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Prism6>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -205,6 +211,15 @@ class Prism6 final : public Prism
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return _n_nodes_on_side<Prism6, 1>(s); }
+
+  /**
+   * \returns 2. Every edge has two nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Prism6>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_pyramid.h b/include/geom/cell_pyramid.h
index 85ad4da710c..ed84c9b55d2 100644
--- a/include/geom/cell_pyramid.h
+++ b/include/geom/cell_pyramid.h
@@ -87,6 +87,9 @@ class Pyramid : public Cell
    */
   virtual unsigned int n_vertices() const override { return 5; }
 
+  virtual unsigned int n_vertices_on_side(const unsigned short s) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 4 - (s != 4); }
+
   /**
    * \returns 8.  All pyramids have 8 edges.
    */
@@ -183,6 +186,18 @@ class Pyramid : public Cell
   static const unsigned int edge_sides_map[8][2];
 
 protected:
+  /**
+   * Helper for n_nodes_on_side() for Pyramid-derived classes.
+   *
+   * That is, returns PyramidClass::nodes_per_side - \p remove_num
+   * for sides 0-3, and PyramidClass::nodes_per_side for side 4.
+  */
+  template <class PyramidClass, unsigned short remove_num>
+  unsigned int _n_nodes_on_side(const unsigned short s) const
+  {
+    static_assert(std::is_base_of<Pyramid, PyramidClass>::value, "Not a Pyramid");
+    return this->_n_nodes_on_side_constant<PyramidClass>(s) - ((s != 4) ? remove_num : 0);
+  }
 
   /**
    * Data for links to parent/neighbor/interior_parent elements.
diff --git a/include/geom/cell_pyramid13.h b/include/geom/cell_pyramid13.h
index 01acc193358..72a7d7e608f 100644
--- a/include/geom/cell_pyramid13.h
+++ b/include/geom/cell_pyramid13.h
@@ -128,8 +128,14 @@ class Pyramid13 final : public Pyramid
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Pyramid13>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Pyramid13>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -233,6 +239,15 @@ class Pyramid13 final : public Pyramid
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return _n_nodes_on_side<Pyramid13, 2>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Pyramid13>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_pyramid14.h b/include/geom/cell_pyramid14.h
index 3a4aae940f8..d009e1f1433 100644
--- a/include/geom/cell_pyramid14.h
+++ b/include/geom/cell_pyramid14.h
@@ -131,8 +131,14 @@ class Pyramid14 final : public Pyramid
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Pyramid14>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Pyramid14>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -254,6 +260,15 @@ class Pyramid14 final : public Pyramid
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return _n_nodes_on_side<Pyramid14, 3>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Pyramid14>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_pyramid5.h b/include/geom/cell_pyramid5.h
index 9c08145b0e8..c4a7272143c 100644
--- a/include/geom/cell_pyramid5.h
+++ b/include/geom/cell_pyramid5.h
@@ -106,8 +106,14 @@ class Pyramid5 final : public Pyramid
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Pyramid5>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Pyramid5>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -198,6 +204,15 @@ class Pyramid5 final : public Pyramid
 
   ElemType side_type (const unsigned int s) const override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return _n_nodes_on_side<Pyramid5, 1>(s); }
+
+  /**
+   * \returns 2. Every edge has two nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Pyramid5>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_tet.h b/include/geom/cell_tet.h
index d9875003e56..a3f2d0c3156 100644
--- a/include/geom/cell_tet.h
+++ b/include/geom/cell_tet.h
@@ -78,6 +78,12 @@ class Tet : public Cell
    */
   virtual unsigned int n_vertices() const override final { return 4; }
 
+  /**
+   * \returns 3. Every side has three vertices.
+   */
+  virtual unsigned int n_vertices_on_side(const unsigned short libmesh_dbg_var(s)) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 3; }
+
   /**
    * \returns 6.  All tetrahedra have 6 edges.
    */
diff --git a/include/geom/cell_tet10.h b/include/geom/cell_tet10.h
index 9bb892f1afb..dbe21e07c2e 100644
--- a/include/geom/cell_tet10.h
+++ b/include/geom/cell_tet10.h
@@ -118,8 +118,14 @@ class Tet10 final : public Tet
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Tet10>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Tet10>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -241,6 +247,18 @@ class Tet10 final : public Tet
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 6. Every side has six nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Tet10>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Tet10>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/cell_tet14.h b/include/geom/cell_tet14.h
index 58ce212430d..2d560170c5c 100644
--- a/include/geom/cell_tet14.h
+++ b/include/geom/cell_tet14.h
@@ -124,8 +124,14 @@ class Tet14 final : public Tet
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Tet14>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Tet14>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -241,6 +247,18 @@ class Tet14 final : public Tet
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 7. Every side has seven nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Tet14>(s); }
+
+  /**
+   * \returns 3. Every edge has three nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Tet14>(e); }
+
 #ifdef LIBMESH_ENABLE_AMR
   virtual
   const std::vector<std::pair<unsigned char, unsigned char>> &
diff --git a/include/geom/cell_tet4.h b/include/geom/cell_tet4.h
index 0ab14cc4122..97b0fc2ee6e 100644
--- a/include/geom/cell_tet4.h
+++ b/include/geom/cell_tet4.h
@@ -113,8 +113,14 @@ class Tet4 final : public Tet
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Tet4>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return this->_nodes_on_edge_ptr<Tet4>(e); }
+
   /**
    * \returns \p true if the specified (local) node number is on the
    * specified edge.
@@ -248,6 +254,18 @@ class Tet4 final : public Tet
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 3. Every side has three nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Tet4>(s); }
+
+  /**
+   * \returns 2. Every edge has two nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->_n_nodes_on_edge_constant<Tet4>(e); }
+
 protected:
 
   /**
diff --git a/include/geom/edge.h b/include/geom/edge.h
index bed5e8e6796..6353ec53eed 100644
--- a/include/geom/edge.h
+++ b/include/geom/edge.h
@@ -58,6 +58,14 @@ class Edge : public Elem
   Edge & operator= (Edge &&) = delete;
   virtual ~Edge() = default;
 
+  /**
+   * Geometric constants for all edges.
+   */
+  static const int num_sides = 2;
+  static const int num_edges = 0;
+  static const int nodes_per_side = 1;
+  static const int nodes_per_edge = invalid_int;
+
   /**
    * \returns 1, the dimensionality of the object.
    */
@@ -78,6 +86,12 @@ class Edge : public Elem
    */
   virtual unsigned int n_vertices() const override final { return 2; }
 
+  /**
+   * \returns 1. Every side has one vertex.
+   */
+  virtual unsigned int n_vertices_on_side(const unsigned short libmesh_dbg_var(s)) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 1; }
+
   /**
    * \returns 0.  All 1D elements have no edges.
    */
@@ -182,11 +196,23 @@ class Edge : public Elem
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { libmesh_assert_less(s, this->n_sides()); return side_nodes_map[s]; }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const override final
+  { return nodes_on_side_ptr(e); }
+
   virtual std::vector<unsigned int> sides_on_edge(const unsigned int) const override final
   { return {}; }
 
+  /**
+   * This maps the \f$ j^{th} \f$ node of the \f$ i^{th} \f$ side to
+   * element node numbers.
+   */
+  static const unsigned int side_nodes_map[2][1];
+
   /**
    * \returns The "circumcenter of mass" (area-weighted average of
    * triangulation circumcenters) of the element.
@@ -208,6 +234,18 @@ class Edge : public Elem
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 1. Every side has one node.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short libmesh_dbg_var(s)) const override final
+  { libmesh_assert_less(s, 2); return 1; }
+
+  /**
+   * The \p Elem::n_nodes_on_edge() member makes no sense for edges.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short) const override final
+  { libmesh_not_implemented(); return 0; }
+
 protected:
 
   /**
diff --git a/include/geom/edge_edge2.h b/include/geom/edge_edge2.h
index 329c9353785..dfcadaa0784 100644
--- a/include/geom/edge_edge2.h
+++ b/include/geom/edge_edge2.h
@@ -171,11 +171,7 @@ class Edge2 : public Edge
    * Geometric constants for Edge2.
    */
   static const int num_nodes = 2;
-  static const int num_sides = 2;
-  static const int num_edges = 0;
   static const int num_children = 2;
-  static const int nodes_per_side = 1;
-  static const int nodes_per_edge = invalid_int;
 
   virtual void flip(BoundaryInfo *) override final;
 
diff --git a/include/geom/edge_edge3.h b/include/geom/edge_edge3.h
index 3ce1285468b..9800444d4f3 100644
--- a/include/geom/edge_edge3.h
+++ b/include/geom/edge_edge3.h
@@ -198,11 +198,7 @@ class Edge3 : public Edge
    * Geometric constants for Edge3.
    */
   static const int num_nodes = 3;
-  static const int num_sides = 2;
-  static const int num_edges = 0;
   static const int num_children = 2;
-  static const int nodes_per_side = 1;
-  static const int nodes_per_edge = invalid_int;
 
   virtual void flip(BoundaryInfo *) override final;
 
diff --git a/include/geom/edge_edge4.h b/include/geom/edge_edge4.h
index 0eccbcb0bb7..eaf692eaa2a 100644
--- a/include/geom/edge_edge4.h
+++ b/include/geom/edge_edge4.h
@@ -192,11 +192,7 @@ class Edge4 final : public Edge
    * Geometric constants for Edge4.
    */
   static const int num_nodes = 4;
-  static const int num_sides = 2;
-  static const int num_edges = 0;
   static const int num_children = 2;
-  static const int nodes_per_side = 1;
-  static const int nodes_per_edge = invalid_int;
 
   virtual void flip(BoundaryInfo *) override final;
 
diff --git a/include/geom/elem.h b/include/geom/elem.h
index 7110fba949e..d99bdb5aeda 100644
--- a/include/geom/elem.h
+++ b/include/geom/elem.h
@@ -39,7 +39,7 @@
 #include <algorithm>
 #include <cstddef>
 #include <iostream>
-#include <limits.h> // CHAR_BIT
+#include <limits.h> // CHAR_BIT, std::numeric_limits
 #include <set>
 #include <vector>
 #include <memory>
@@ -224,18 +224,13 @@ class Elem : public ReferenceCountedObject<Elem>,
   /**
    * A static integral constant representing an invalid subdomain id.
    * See also DofObject::{invalid_id, invalid_unique_id, invalid_processor_id}.
-   *
-   * \note We don't use the static_cast(-1) trick here since
-   * \p subdomain_id_type is sometimes a *signed* integer for
-   * compatibility reasons (see libmesh/id_types.h).
-   *
-   * \note Normally you can declare static const integral types
-   * directly in the header file (C++ standard, 9.4.2/4) but
-   * std::numeric_limits<T>::max() is not considered a "constant
-   * expression".  This one is therefore defined in elem.C.
-   * http://stackoverflow.com/questions/2738435/using-numeric-limitsmax-in-constant-expressions
    */
-  static const subdomain_id_type invalid_subdomain_id;
+  static constexpr subdomain_id_type invalid_subdomain_id = std::numeric_limits<subdomain_id_type>::max();
+
+  /**
+   * A static integral constant representing an invalid index to a vertex.
+  */
+  static constexpr unsigned short invalid_vertex = std::numeric_limits<unsigned short>::max();
 
   /**
    * \returns A pointer to the "reference element" associated
@@ -598,6 +593,11 @@ class Elem : public ReferenceCountedObject<Elem>,
    */
   virtual unsigned int n_nodes () const = 0;
 
+  /**
+   * \returns The number of nodes on the side with index \p s.
+   */
+  virtual unsigned int n_nodes_on_side (const unsigned short s) const = 0;
+
   /**
    * The maximum number of nodes *any* element can contain.
    * This is useful for replacing heap vectors with stack arrays.
@@ -642,6 +642,12 @@ class Elem : public ReferenceCountedObject<Elem>,
    */
   IntRange<unsigned short> side_index_range () const;
 
+  /**
+   * \returns An integer range from 0 up to (but not including)
+   * the number of vertices this element has.
+   */
+  IntRange<unsigned short> vertex_index_range () const;
+
   /**
    * \returns The number of neighbors the element that has been derived
    * from this class has.
@@ -660,12 +666,22 @@ class Elem : public ReferenceCountedObject<Elem>,
    */
   virtual unsigned int n_vertices () const = 0;
 
+  /**
+   * \returns The number of verticies on the side with index \p s.
+   */
+  virtual unsigned int n_vertices_on_side (const unsigned short s) const = 0;
+
   /**
    * \returns The number of edges the element that has been derived
    * from this class has.
    */
   virtual unsigned int n_edges () const = 0;
 
+  /**
+   * \returns The number of nodes on the edge with index \p e.
+   */
+  virtual unsigned int n_nodes_on_edge (const unsigned short) const = 0;
+
   /**
    * \returns An integer range from 0 up to (but not including)
    * the number of edges this element has.
@@ -734,11 +750,29 @@ class Elem : public ReferenceCountedObject<Elem>,
    */
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int /*s*/) const = 0;
 
+  /**
+   * \returns A pointer to the beginning of the array that contains the
+   * (local) node numbers on the side \p s
+   *
+   * This does not create temporaries, unlike \p nodes_on_side.
+   * Use n_nodes_on_side for indexing.
+   */
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const = 0;
+
   /**
    * \returns the (local) node numbers on the specified edge
    */
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int /*e*/) const = 0;
 
+  /**
+   * \returns A pointer to the beginning of the array that contains the
+   * (local) node numbers on the edge \p e
+   *
+   * This does not create temporaries, unlike \p nodes_on_edge.
+   * Use n_nodes_on_edge for indexing.
+   */
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short e) const = 0;
+
   /**
    * \returns the (local) side numbers that touch the specified edge
    */
@@ -1989,6 +2023,55 @@ class Elem : public ReferenceCountedObject<Elem>,
                              const unsigned int i,
                              ElemType edgetype);
 
+  /**
+   * Helper for overriding n_nodes_on_side in derived classes.
+   *
+   * Performs bound checking on \p s and returns nodes_per_side
+   * for the given class for all sides.
+  */
+  template <class ElemClass>
+  unsigned int _n_nodes_on_side_constant(const unsigned short libmesh_dbg_var(s)) const
+  {
+    static_assert(std::is_base_of<Elem, ElemClass>::value, "Not an Elem");
+    libmesh_assert_less(s, this->n_sides());
+    return ElemClass::nodes_per_side;
+  }
+
+  /**
+   * Helper for overriding n_nodes_per_edge in derived classes.
+   *
+   * Performs bound checking on \p s and returns nodes_per_edge
+   * for the given class for all edges.
+  */
+  template <class ElemClass>
+  unsigned int _n_nodes_on_edge_constant(const unsigned short libmesh_dbg_var(s)) const
+  {
+    static_assert(std::is_base_of<Elem, ElemClass>::value, "Not an Elem");
+    libmesh_assert_less(s, this->n_edges());
+    return ElemClass::nodes_per_edge;
+  }
+
+  /**
+   * Helper for overriding nodes_on_side in derived classes.
+  */
+  template <class ElemClass>
+  const unsigned int * _nodes_on_side_ptr(const unsigned short s) const
+  {
+    static_assert(std::is_base_of<Elem, ElemClass>::value, "Not an Elem");
+    libmesh_assert_less(s, std::extent<decltype(ElemClass::side_nodes_map)>::value);
+    return ElemClass::side_nodes_map[s];
+  }
+
+  /**
+   * Helper for overriding nodes_on_side in derived classes.
+  */
+  template <class ElemClass>
+  const unsigned int * _nodes_on_edge_ptr(const unsigned short s) const
+  {
+    static_assert(std::is_base_of<Elem, ElemClass>::value, "Not an Elem");
+    libmesh_assert_less(s, std::extent<decltype(ElemClass::edge_nodes_map)>::value);
+    return ElemClass::edge_nodes_map[s];
+  }
 
 #ifdef LIBMESH_ENABLE_AMR
 
@@ -2465,6 +2548,13 @@ Elem::side_index_range() const
 
 
 
+inline
+IntRange<unsigned short>
+Elem::vertex_index_range() const
+{
+  return {0, cast_int<unsigned short>(this->n_vertices())};
+}
+
 
 inline
 std::unique_ptr<const Elem> Elem::side_ptr (unsigned int i) const
diff --git a/include/geom/elem_corner.h b/include/geom/elem_corner.h
new file mode 100644
index 00000000000..1e070bc7ec3
--- /dev/null
+++ b/include/geom/elem_corner.h
@@ -0,0 +1,188 @@
+// The libMesh Finite Element Library.
+// Copyright (C) 2002-2022 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
+
+// This library is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+// Lesser General Public License for more details.
+
+// You should have received a copy of the GNU Lesser General Public
+// License along with this library; if not, write to the Free Software
+// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+
+
+
+#ifndef LIBMESH_ELEMCORNER_H
+#define LIBMESH_ELEMCORNER_H
+
+#include "libmesh/libmesh_common.h"
+
+#include "libmesh/elem.h"
+
+#include <utility> // std::pair
+
+namespace libMesh
+{
+
+/**
+ * The \p ElemCorner is a helper class for defining whether or not
+ * _something_ is at an element vertex or edge. In this context,
+ * we define corner as a vertex in 2D and as a vertex or an edge
+ * in 3D.
+ *
+ * This class is typically used in intersection algorithms, in which
+ * it should be interfaced with by using the \p at_corner(),
+ * \p at_edge(), and \p at_vertex() methods.
+ *
+ * Internally, it keeps track of whether or not at one of these
+ * "corners" by storing a set of indicies that denote vertex indices.
+ * If the first is valid and the second is not, it implies
+ * being at the vertex with said index. If both are valid, it implies
+ * being at the edge defined by the verticies with said indices.
+ */
+class ElemCorner : public std::pair<unsigned short, unsigned short>
+{
+public:
+  /**
+   * Default constructor: sets entires to invalid (not at vertex or edge)
+   */
+  ElemCorner()
+    : std::pair<unsigned short, unsigned short>(Elem::invalid_vertex,
+                                                Elem::invalid_vertex) { }
+
+  /**
+   * Constructor with given vertex indices \p v1 and \p v2.
+   */
+  ElemCorner(const unsigned short v1, const unsigned short v2)
+    : std::pair<unsigned short, unsigned short>(v1, v2) { }
+
+  /**
+   * @returns true if at the corner (edge or vertex)
+   */
+  bool at_corner() const { return first != Elem::invalid_vertex; }
+
+  /**
+   * @returns true if the vertex index data is invalid
+   */
+  bool is_invalid() const
+  { return first == Elem::invalid_vertex && second == Elem::invalid_vertex; }
+
+  /**
+   * @returns true if at a vertex
+   */
+  bool at_vertex() const
+  { return first != Elem::invalid_vertex && second == Elem::invalid_vertex; }
+  /**
+   * @returns true if at vertex \p v
+   */
+  bool at_vertex(const unsigned short v) const
+  { return first == v && second == Elem::invalid_vertex; }
+
+  /**
+   * Invalidates the current state
+   */
+  void invalidate()
+  {
+    first = Elem::invalid_vertex;
+    second = Elem::invalid_vertex;
+  }
+
+  /**
+   * @returns The vertex ID when at a vertex
+   */
+  unsigned short vertex() const
+  { libmesh_assert(at_vertex()); return first; }
+
+  /**
+   * Prints the current state (at edge, at vertex, not at either)
+   */
+  std::string print() const;
+
+  /**
+   * Sets the "at vertex" state
+   */
+  void set_vertex(const unsigned short v)
+  {
+    libmesh_assert_not_equal_to(v, Elem::invalid_vertex);
+    first = v;
+    second = Elem::invalid_vertex;
+  }
+
+  /**
+   * @returns true if at an edge
+   */
+  bool at_edge() const
+  { return first != Elem::invalid_vertex && second != Elem::invalid_vertex; }
+  /**
+   * @returns true if at the edge defined by vertices \p v1 and \p v2
+   */
+  bool at_edge(const unsigned short v1, const unsigned short v2) const
+  {
+    libmesh_assert_not_equal_to(v1, Elem::invalid_vertex);
+    libmesh_assert_not_equal_to(v2, Elem::invalid_vertex);
+    libmesh_assert_not_equal_to(v1, v2);
+    return (first == v1 && second == v2) || (first == v2 && second == v1);
+  }
+  /**
+   * @returns true if at the edge with index \p e on element \p
+   */
+  bool at_edge(const Elem & elem, const unsigned short e) const;
+
+  /**
+   * @returns The vertices that contain the edge when at an edge
+   */
+  const std::pair<unsigned short, unsigned short> & edge_vertices() const
+  { libmesh_assert(at_edge()); return *this; }
+
+  /**
+   * Sets the "at edge" state
+   */
+  void set_edge(const unsigned short v1, const unsigned short v2)
+  {
+    libmesh_assert_not_equal_to(v1, Elem::invalid_vertex);
+    libmesh_assert_not_equal_to(v2, Elem::invalid_vertex);
+    libmesh_assert_not_equal_to(v1, v2);
+    first = v1;
+    second = v2;
+  }
+  /**
+   * Sets the "at edge" state
+   */
+  void set_edge(const std::pair<unsigned short, unsigned short> & vs)
+  { set_edge(vs.first, vs.second); }
+
+  /**
+   * Builds an edge of element \p elem into the Elem \p edge when at an edge
+   */
+  void build_edge(const Elem & elem, std::unique_ptr<const Elem> & edge) const;
+  /**
+   * @returns The built edge of element \p elem when at an edge
+   */
+  std::unique_ptr<const Elem> build_edge(const Elem & elem) const;
+
+  /**
+   * @returns The vertex point when at a vertex
+   */
+  const Point & vertex_point(const Elem & elem) const
+  { return elem.point(vertex()); }
+
+  /**
+   * @returns Whether or not the current state (at vertex/edge) is valid
+   * for the given \p elem and \p point. \p tol is used as the tolerance
+   * to pass to the equality checks.
+   *
+   * This ONLY checks for validity when at_corner().
+   */
+  bool is_valid(const Elem & elem, const Point & point, const Real tol = TOLERANCE) const;
+};
+
+} // namespace libMesh
+
+std::ostream & operator<<(std::ostream & os, const libMesh::ElemCorner & elem_corner);
+
+#endif // LIBMESH_ELEMCORNER_H
diff --git a/include/geom/face.h b/include/geom/face.h
index d3a818bf217..3b7176e41ca 100644
--- a/include/geom/face.h
+++ b/include/geom/face.h
@@ -66,6 +66,18 @@ class Face : public Elem
    */
   virtual unsigned int n_faces() const override final { return 0; }
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short s) const override final
+  { return nodes_on_side_ptr(s); }
+
+  /**
+   * \returns 2. Every side has two vertices.
+   */
+  virtual unsigned int n_vertices_on_side(const unsigned short libmesh_dbg_var(s)) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 2; }
+
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->n_nodes_on_side(e); }
+
   /**
    * build_side and build_edge are identical for faces.
    */
diff --git a/include/geom/face_inf_quad.h b/include/geom/face_inf_quad.h
index 04fc4c3b78e..022ba8da8c5 100644
--- a/include/geom/face_inf_quad.h
+++ b/include/geom/face_inf_quad.h
@@ -106,6 +106,12 @@ class InfQuad : public Elem
    */
   virtual unsigned int n_vertices() const override final { return 4; }
 
+  /**
+   * \returns 2. Every side has two vertices.
+   */
+  virtual unsigned int n_vertices_on_side(const unsigned short libmesh_dbg_var(s)) const override final
+  { libmesh_assert_less(s, this->n_sides()); return 2; }
+
   /**
    * \returns 3.  All infinite quads have 1 edge in the
    * base, and 2 perpendicular to the base.
@@ -233,6 +239,12 @@ class InfQuad : public Elem
 
   virtual void orient(BoundaryInfo *) override final;
 
+  virtual unsigned int n_nodes_on_edge(const unsigned short e) const override final
+  { return this->n_nodes_on_side(e); }
+
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short s) const override final
+  { return nodes_on_side_ptr(s); }
+
 protected:
 
   /**
diff --git a/include/geom/face_inf_quad4.h b/include/geom/face_inf_quad4.h
index 1500d521df1..6f7fc18fdf7 100644
--- a/include/geom/face_inf_quad4.h
+++ b/include/geom/face_inf_quad4.h
@@ -105,6 +105,9 @@ class InfQuad4 : public InfQuad
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<InfQuad4>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
   /**
@@ -164,6 +167,9 @@ class InfQuad4 : public InfQuad
 
   virtual void flip(BoundaryInfo *) override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<InfQuad4>(s); }
+
 protected:
 
   /**
diff --git a/include/geom/face_inf_quad6.h b/include/geom/face_inf_quad6.h
index 28d36d38dd8..559a27d06f3 100644
--- a/include/geom/face_inf_quad6.h
+++ b/include/geom/face_inf_quad6.h
@@ -106,6 +106,9 @@ class InfQuad6 : public InfQuad
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<InfQuad6>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
   /**
@@ -200,6 +203,9 @@ class InfQuad6 : public InfQuad
 
   virtual void flip(BoundaryInfo *) override final;
 
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<InfQuad6>(s) - (s == 0 ? 0 : 1); }
+
 protected:
 
   /**
diff --git a/include/geom/face_quad4.h b/include/geom/face_quad4.h
index c2c44ae1e06..c032f03a935 100644
--- a/include/geom/face_quad4.h
+++ b/include/geom/face_quad4.h
@@ -101,6 +101,9 @@ class Quad4 : public Quad
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Quad4>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
   /**
@@ -178,6 +181,12 @@ class Quad4 : public Quad
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 2. Every side has seven nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Quad4>(s); }
+
 protected:
 
   /**
diff --git a/include/geom/face_quad8.h b/include/geom/face_quad8.h
index 731388b7a86..3ce33cb8411 100644
--- a/include/geom/face_quad8.h
+++ b/include/geom/face_quad8.h
@@ -104,6 +104,9 @@ class Quad8 : public Quad
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Quad8>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
   /**
@@ -217,6 +220,12 @@ class Quad8 : public Quad
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 3. Every side has three nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Quad8>(s); }
+
 protected:
 
   /**
diff --git a/include/geom/face_quad9.h b/include/geom/face_quad9.h
index 33f5c440e62..8edbb54f5cf 100644
--- a/include/geom/face_quad9.h
+++ b/include/geom/face_quad9.h
@@ -104,6 +104,9 @@ class Quad9 : public Quad
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Quad9>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
   /**
@@ -224,6 +227,12 @@ class Quad9 : public Quad
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 3. Every side has three nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Quad9>(s); }
+
 protected:
 
   /**
diff --git a/include/geom/face_tri3.h b/include/geom/face_tri3.h
index 132e4c0078e..5d9d39c7fa3 100644
--- a/include/geom/face_tri3.h
+++ b/include/geom/face_tri3.h
@@ -109,6 +109,9 @@ class Tri3 : public Tri
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Tri3>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
   /**
@@ -203,6 +206,12 @@ class Tri3 : public Tri
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 2. Every side has two nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Tri3>(s); }
+
 protected:
 
   /**
diff --git a/include/geom/face_tri6.h b/include/geom/face_tri6.h
index 2d568fa90aa..900fa09fdbc 100644
--- a/include/geom/face_tri6.h
+++ b/include/geom/face_tri6.h
@@ -114,6 +114,9 @@ class Tri6 : public Tri
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Tri6>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
   /**
@@ -227,6 +230,12 @@ class Tri6 : public Tri
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 3. Every side has three nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Tri6>(s); }
+
 protected:
 
   /**
diff --git a/include/geom/face_tri7.h b/include/geom/face_tri7.h
index 953896d2d13..589eb7de242 100644
--- a/include/geom/face_tri7.h
+++ b/include/geom/face_tri7.h
@@ -114,6 +114,9 @@ class Tri7 : public Tri
 
   virtual std::vector<unsigned int> nodes_on_side(const unsigned int s) const override;
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short s) const override final
+  { return this->_nodes_on_side_ptr<Tri7>(s); }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int e) const override;
 
   /**
@@ -233,6 +236,12 @@ class Tri7 : public Tri
 
   ElemType side_type (const unsigned int s) const override final;
 
+  /**
+   * \returns 3. Every side has three nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short s) const override final
+  { return this->_n_nodes_on_side_constant<Tri7>(s); }
+
 protected:
 
   /**
diff --git a/include/geom/intersection_tools.h b/include/geom/intersection_tools.h
new file mode 100644
index 00000000000..2ccba1e93c7
--- /dev/null
+++ b/include/geom/intersection_tools.h
@@ -0,0 +1,165 @@
+// The libMesh Finite Element Library.
+// Copyright (C) 2002-2022 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
+
+// This library is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+// Lesser General Public License for more details.
+
+// You should have received a copy of the GNU Lesser General Public
+// License along with this library; if not, write to the Free Software
+// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+
+
+
+#ifndef LIBMESH_INTERSECTIONTOOLS_H
+#define LIBMESH_INTERSECTIONTOOLS_H
+
+#include "libmesh/libmesh_common.h"
+
+namespace libMesh
+{
+
+class Elem;
+class ElemCorner;
+class Point;
+
+namespace IntersectionTools
+{
+
+/**
+ * Enum that represents the result of the \p within_segment() methods.
+*/
+enum WithinSegmentResult : int {
+                         NOT_WITHIN = 0,
+                         AT_BEGINNING = 1,
+                         AT_END = 2,
+                         BETWEEN = 3 };
+
+/**
+ * Checks whether or not a point is within a line segment
+ * @param s1 The first point on the segment
+ * @param s2 The second point on the segment
+ * @param p The point
+ * @param tol The relative tolerance to use
+ * @return Enum denoting whether or not the point is not within, at s1,
+ * at s2, or between [s1, s2]
+ */
+WithinSegmentResult within_segment(const Point & s1,
+                                   const Point & s2,
+                                   const Point & p,
+                                   const Real tol = TOLERANCE);
+
+/**
+ * Checks whether or not the given points are collinear
+ * @param p1 The first point
+ * @param p2 The second point
+ * @param p3 The third point
+ * @param tol The tolerance to use (absolute tolerance of the cosine
+ * of the angle between [p1 -> p2] and [p1 -> p3])
+ * @return Whether or not the given points are collinear
+ */
+bool collinear(const Point & p1,
+               const Point & p2,
+               const Point & p3,
+               const Real tol = TOLERANCE);
+
+/**
+ * @returns Whether or not the edges on element \p element are
+ * individually collinear.
+*/
+bool edges_are_collinear(const Elem & elem, const Real tol = TOLERANCE);
+
+/**
+ * \returns True if the given point is contained within an edge an element
+ * @param elem The element
+ * @param p The point
+ * @param corner To be filled with the edge/vertex that the point
+ * is within/at or, if any (must be initially invalid)
+ * @param linearize Whether or not to "linearize" the check, if this
+ * is set to false and edges are found to not be collinear, an error
+ * is thrown
+ * @param tol Relative tolerance to use in comparison
+ *
+ * \p corner will be set to an "at vertex" state if the point is
+ * both within the edge _and_ at a vertex.
+*/
+bool within_edge(const Elem & elem,
+                 const Point & p,
+                 ElemCorner & corner,
+                 const bool linearize = false,
+                 const Real tol = TOLERANCE);
+
+/**
+ * \returns True if the given point is contained within an edge
+ * on the given side of an element
+ * @param elem The element
+ * @param p The point
+ * @param s The side
+ * @param corner To be filled with the edge/vertex that the point
+ * is within/at or, if any (must be initially invalid)
+ * @param linearize Whether or not to "linearize" the check, if this
+ * is set to false and edges are found to not be collinear, an error
+ * is thrown
+ * @param tol Relative tolerance to use in comparison
+ *
+ * \p corner will be set to an "at vertex" state if the point is
+ * both within the edge _and_ at a vertex.
+ *
+ * This method is only implemented for three-dimensional, finite
+ * elements.
+*/
+bool within_edge_on_side(const Elem & elem,
+                         const Point & p,
+                         const unsigned short s,
+                         ElemCorner & corner,
+                         const bool linearize = false,
+                         const Real tol = TOLERANCE);
+
+/**
+ * \returns The index of the vertex that the given point is at
+ * on the given element, invalid otherwise
+ * @param elem The element
+ * @param p The point
+ * @param tol The relative tolerance used in point comparison
+*/
+unsigned short at_vertex(const Elem & elem,
+                         const Point & p,
+                         const Real tol = TOLERANCE);
+
+/**
+ * \returns The index of the vertex that the given point is at
+ * on the given element on the given side, invalid otherwise
+ * @param elem The element
+ * @param p The point
+ * @param s The side
+ * @param tol The relative tolerance used in point comparison
+*/
+unsigned short at_vertex_on_side(const Elem & elem,
+                                 const Point & p,
+                                 const unsigned short s,
+                                 const Real tol = TOLERANCE);
+
+namespace detail
+{
+/**
+ * Internal method for checking whether or not the point \p
+ * is within the edge defined by vertices \p v1 and \p v2
+ * on element \p elem.
+*/
+bool _within_edge(const Elem & elem,
+                  const Point & p,
+                  ElemCorner & corner,
+                  const unsigned int v1,
+                  const unsigned int v2,
+                  const Real tol);
+} // namespace detail
+} // namespace IntersectionTools
+} // namespace libMesh
+
+#endif // LIBMESH_INTERSECTIONTOOLS_H
diff --git a/include/geom/node_elem.h b/include/geom/node_elem.h
index ea8724afda4..7bd02dac186 100644
--- a/include/geom/node_elem.h
+++ b/include/geom/node_elem.h
@@ -77,6 +77,30 @@ class NodeElem : public Elem
    */
   virtual unsigned int n_nodes() const override { return 1; }
 
+  /**
+   * The \p Elem::n_nodes_on_side makes no sense for nodes.
+   */
+  virtual unsigned int n_nodes_on_side(const unsigned short) const override
+  { libmesh_not_implemented(); return 0; }
+
+  /**
+   * The \p Elem::n_nodes_on_edge makes no sense for nodes.
+   */
+  virtual unsigned int n_nodes_on_edge(const unsigned short) const override
+  { libmesh_not_implemented(); return 0; }
+
+  /**
+   * The \p Elem::nodes_on_side_ptr makes no sense for nodes.
+   */
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short) const override final
+  { libmesh_not_implemented(); return nullptr; }
+
+  /**
+   * The \p Elem::nodes_on_edge_ptr makes no sense for nodes.
+   */
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short) const override final
+  { libmesh_not_implemented(); return nullptr; }
+
   /**
    * \returns 0.
    */
@@ -87,6 +111,12 @@ class NodeElem : public Elem
    */
   virtual unsigned int n_vertices() const override { return 1; }
 
+  /**
+   * The \p Elem::n_vertices_on_side makes no sense for nodes.
+   */
+  virtual unsigned int n_vertices_on_side(const unsigned short) const override
+  { libmesh_not_implemented(); return 0; }
+
   /**
    * \returns 0.
    */
diff --git a/include/geom/remote_elem.h b/include/geom/remote_elem.h
index 5f8267843bf..7e2f8066946 100644
--- a/include/geom/remote_elem.h
+++ b/include/geom/remote_elem.h
@@ -119,12 +119,21 @@ class RemoteElem : public Elem,
   virtual unsigned int n_nodes () const override
   { libmesh_not_implemented(); return 0; }
 
+  virtual unsigned int n_nodes_on_side (const unsigned short) const override
+  { libmesh_not_implemented(); return 0; }
+
+  virtual unsigned int n_nodes_on_edge (const unsigned short) const override
+  { libmesh_not_implemented(); return 0; }
+
   virtual unsigned int n_sides () const override
   { libmesh_not_implemented(); return 0; }
 
   virtual unsigned int n_vertices () const override
   { libmesh_not_implemented(); return 0; }
 
+  virtual unsigned int n_vertices_on_side(const unsigned short) const override
+  { libmesh_not_implemented(); return 0; }
+
   virtual unsigned int n_edges () const override
   { libmesh_not_implemented(); return 0; }
 
@@ -153,12 +162,18 @@ class RemoteElem : public Elem,
     return {0};
   }
 
+  virtual const unsigned int * nodes_on_side_ptr(const unsigned short) const override
+  { libmesh_not_implemented(); return nullptr; }
+
   virtual std::vector<unsigned int> nodes_on_edge(const unsigned int) const override
   {
     libmesh_not_implemented();
     return {0};
   }
 
+  virtual const unsigned int * nodes_on_edge_ptr(const unsigned short) const override
+  { libmesh_not_implemented(); return nullptr; }
+
   virtual std::vector<unsigned int> sides_on_edge(const unsigned int) const override
   {
     libmesh_not_implemented();
diff --git a/include/include_HEADERS b/include/include_HEADERS
index 85b43d9d4f1..6f2a7677613 100644
--- a/include/include_HEADERS
+++ b/include/include_HEADERS
@@ -139,6 +139,7 @@ include_HEADERS =  \
         geom/edge_edge4.h \
         geom/edge_inf_edge2.h \
         geom/elem.h \
+        geom/elem_corner.h \
         geom/elem_cutter.h \
         geom/elem_hash.h \
         geom/elem_internal.h \
@@ -161,6 +162,7 @@ include_HEADERS =  \
         geom/face_tri3_subdivision.h \
         geom/face_tri6.h \
         geom/face_tri7.h \
+        geom/intersection_tools.h \
         geom/node.h \
         geom/node_elem.h \
         geom/node_range.h \
diff --git a/include/libmesh/Makefile.am b/include/libmesh/Makefile.am
index 2d49e348066..6f0f85b51ad 100644
--- a/include/libmesh/Makefile.am
+++ b/include/libmesh/Makefile.am
@@ -129,6 +129,7 @@ BUILT_SOURCES = \
         edge_edge4.h \
         edge_inf_edge2.h \
         elem.h \
+        elem_corner.h \
         elem_cutter.h \
         elem_hash.h \
         elem_internal.h \
@@ -151,6 +152,7 @@ BUILT_SOURCES = \
         face_tri3_subdivision.h \
         face_tri6.h \
         face_tri7.h \
+        intersection_tools.h \
         node.h \
         node_elem.h \
         node_range.h \
@@ -947,6 +949,9 @@ edge_inf_edge2.h: $(top_srcdir)/include/geom/edge_inf_edge2.h
 elem.h: $(top_srcdir)/include/geom/elem.h
 	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
 
+elem_corner.h: $(top_srcdir)/include/geom/elem_corner.h
+	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
+
 elem_cutter.h: $(top_srcdir)/include/geom/elem_cutter.h
 	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
 
@@ -1013,6 +1018,9 @@ face_tri6.h: $(top_srcdir)/include/geom/face_tri6.h
 face_tri7.h: $(top_srcdir)/include/geom/face_tri7.h
 	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
 
+intersection_tools.h: $(top_srcdir)/include/geom/intersection_tools.h
+	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
+
 node.h: $(top_srcdir)/include/geom/node.h
 	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
 
diff --git a/include/libmesh/Makefile.in b/include/libmesh/Makefile.in
index b1b04d126ea..032f07d7980 100644
--- a/include/libmesh/Makefile.in
+++ b/include/libmesh/Makefile.in
@@ -547,15 +547,16 @@ BUILT_SOURCES = auto_ptr.h default_coupling.h dirichlet_boundaries.h \
 	cell_pyramid5.h cell_tet.h cell_tet10.h cell_tet14.h \
 	cell_tet4.h compare_elems_by_level.h edge.h edge_edge2.h \
 	edge_edge3.h edge_edge4.h edge_inf_edge2.h elem.h \
-	elem_cutter.h elem_hash.h elem_internal.h elem_quality.h \
-	elem_range.h elem_side_builder.h face.h face_inf_quad.h \
-	face_inf_quad4.h face_inf_quad6.h face_quad.h face_quad4.h \
-	face_quad4_shell.h face_quad8.h face_quad8_shell.h \
-	face_quad9.h face_tri.h face_tri3.h face_tri3_shell.h \
-	face_tri3_subdivision.h face_tri6.h face_tri7.h node.h \
-	node_elem.h node_range.h plane.h point.h reference_elem.h \
-	remote_elem.h side.h sphere.h stored_range.h surface.h \
-	abaqus_io.h boundary_info.h boundary_mesh.h checkpoint_io.h \
+	elem_corner.h elem_cutter.h elem_hash.h elem_internal.h \
+	elem_quality.h elem_range.h elem_side_builder.h face.h \
+	face_inf_quad.h face_inf_quad4.h face_inf_quad6.h face_quad.h \
+	face_quad4.h face_quad4_shell.h face_quad8.h \
+	face_quad8_shell.h face_quad9.h face_tri.h face_tri3.h \
+	face_tri3_shell.h face_tri3_subdivision.h face_tri6.h \
+	face_tri7.h intersection_tools.h node.h node_elem.h \
+	node_range.h plane.h point.h reference_elem.h remote_elem.h \
+	side.h sphere.h stored_range.h surface.h abaqus_io.h \
+	boundary_info.h boundary_mesh.h checkpoint_io.h \
 	distributed_mesh.h dyna_io.h ensight_io.h exodusII_io.h \
 	exodusII_io_helper.h exodus_header_info.h fro_io.h gmsh_io.h \
 	gmv_io.h gnuplot_io.h inf_elem_builder.h matlab_io.h \
@@ -1282,6 +1283,9 @@ edge_inf_edge2.h: $(top_srcdir)/include/geom/edge_inf_edge2.h
 elem.h: $(top_srcdir)/include/geom/elem.h
 	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
 
+elem_corner.h: $(top_srcdir)/include/geom/elem_corner.h
+	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
+
 elem_cutter.h: $(top_srcdir)/include/geom/elem_cutter.h
 	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
 
@@ -1348,6 +1352,9 @@ face_tri6.h: $(top_srcdir)/include/geom/face_tri6.h
 face_tri7.h: $(top_srcdir)/include/geom/face_tri7.h
 	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
 
+intersection_tools.h: $(top_srcdir)/include/geom/intersection_tools.h
+	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
+
 node.h: $(top_srcdir)/include/geom/node.h
 	$(AM_V_GEN)rm -f $@ && $(LN_S) -f $< $@
 
diff --git a/include/utils/xdr_cxx.h b/include/utils/xdr_cxx.h
index 1a932784e19..15671ab501c 100644
--- a/include/utils/xdr_cxx.h
+++ b/include/utils/xdr_cxx.h
@@ -40,6 +40,7 @@
 #include <iosfwd>
 #include <vector>
 #include <string>
+#include <string_view>
 
 const unsigned int xdr_MAX_STRING_LENGTH=256;
 
@@ -128,7 +129,7 @@ class Xdr
    * Inputs or outputs a single value.
    */
   template <typename T>
-  void data(T & a, const char * comment="");
+  void data(T & a, std::string_view comment="");
 
   /**
    * Same, but provides an \p ostream like interface.
diff --git a/src/geom/edge.C b/src/geom/edge.C
index ebe238dfbdc..a8400cf9dd6 100644
--- a/src/geom/edge.C
+++ b/src/geom/edge.C
@@ -24,6 +24,21 @@
 namespace libMesh
 {
 
+
+// ------------------------------------------------------------
+// Edge class static member initializations
+
+const int Edge::num_sides;
+const int Edge::num_edges;
+const int Edge::nodes_per_side;
+const int Edge::nodes_per_edge;
+
+const unsigned int Edge::side_nodes_map[Edge::num_sides][Edge::nodes_per_side] =
+  {
+    {0}, // Side 0
+    {1}, // Side 1
+  };
+
 unsigned int Edge::local_side_node(unsigned int side,
                                    unsigned int /*side_node*/) const
 {
diff --git a/src/geom/edge_edge2.C b/src/geom/edge_edge2.C
index b8650e0b653..671294b123a 100644
--- a/src/geom/edge_edge2.C
+++ b/src/geom/edge_edge2.C
@@ -28,11 +28,7 @@ namespace libMesh
 
 // Edge2 class static member initializations
 const int Edge2::num_nodes;
-const int Edge2::num_sides;
-const int Edge2::num_edges;
 const int Edge2::num_children;
-const int Edge2::nodes_per_side;
-const int Edge2::nodes_per_edge;
 
 #ifdef LIBMESH_ENABLE_AMR
 
diff --git a/src/geom/edge_edge3.C b/src/geom/edge_edge3.C
index f99d691d070..e48b8ddb8dc 100644
--- a/src/geom/edge_edge3.C
+++ b/src/geom/edge_edge3.C
@@ -27,11 +27,7 @@ namespace libMesh
 
 // Edge3 class static member initializations
 const int Edge3::num_nodes;
-const int Edge3::num_sides;
-const int Edge3::num_edges;
 const int Edge3::num_children;
-const int Edge3::nodes_per_side;
-const int Edge3::nodes_per_edge;
 
 #ifdef LIBMESH_ENABLE_AMR
 
diff --git a/src/geom/edge_edge4.C b/src/geom/edge_edge4.C
index e23a69e1963..25ea0b12ea1 100644
--- a/src/geom/edge_edge4.C
+++ b/src/geom/edge_edge4.C
@@ -27,11 +27,7 @@ namespace libMesh
 
 // Edge4 class static member initializations
 const int Edge4::num_nodes;
-const int Edge4::num_sides;
-const int Edge4::num_edges;
 const int Edge4::num_children;
-const int Edge4::nodes_per_side;
-const int Edge4::nodes_per_edge;
 
 #ifdef LIBMESH_ENABLE_AMR
 
diff --git a/src/geom/elem.C b/src/geom/elem.C
index bec33a4a2ee..efad9187086 100644
--- a/src/geom/elem.C
+++ b/src/geom/elem.C
@@ -82,7 +82,6 @@
 #include <array>
 #include <iterator>  // for std::ostream_iterator
 #include <sstream>
-#include <limits>    // for std::numeric_limits<>
 #include <cmath>     // for std::sqrt()
 #include <memory>
 
@@ -93,8 +92,6 @@ namespace libMesh
 Threads::spin_mutex parent_indices_mutex;
 Threads::spin_mutex parent_bracketing_nodes_mutex;
 
-const subdomain_id_type Elem::invalid_subdomain_id = std::numeric_limits<subdomain_id_type>::max();
-
 // Initialize static member variables
 const unsigned int Elem::max_n_nodes;
 
diff --git a/src/geom/elem_corner.C b/src/geom/elem_corner.C
new file mode 100644
index 00000000000..9468449e6f5
--- /dev/null
+++ b/src/geom/elem_corner.C
@@ -0,0 +1,92 @@
+// The libMesh Finite Element Library.
+// Copyright (C) 2002-2022 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
+
+// This library is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+// Lesser General Public License for more details.
+
+// You should have received a copy of the GNU Lesser General Public
+// License along with this library; if not, write to the Free Software
+// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+
+
+// Local includes
+#include "libmesh/elem_corner.h"
+
+
+namespace libMesh
+{
+
+bool ElemCorner::at_edge(const Elem & elem, const unsigned short e) const
+{
+  libmesh_assert_greater(elem.dim(), 1);
+  libmesh_assert_less(e, elem.n_edges());
+  if (!at_edge())
+    return false;
+  const unsigned int * nodes_on_edge_ptr = elem.nodes_on_edge_ptr(e);
+  return at_edge(nodes_on_edge_ptr[0], nodes_on_edge_ptr[1]);
+}
+
+void ElemCorner::build_edge(const Elem & elem, std::unique_ptr<const Elem> & edge) const
+{
+  libmesh_assert_greater(elem.dim(), 1);
+  libmesh_assert(at_edge());
+  libmesh_assert_less(first, elem.n_vertices());
+  libmesh_assert_less(second, elem.n_vertices());
+
+  for (const auto e : elem.edge_index_range())
+    if (elem.is_node_on_edge(first, e) && elem.is_node_on_edge(second, e))
+    {
+      elem.build_edge_ptr(edge, e);
+      return;
+    }
+
+  libmesh_error_msg("Element does not contain vertices in ElemCorner");
+}
+
+std::unique_ptr<const Elem> ElemCorner::build_edge(const Elem & elem) const
+{
+  std::unique_ptr<const Elem> edge;
+  build_edge(elem, edge);
+  return edge;
+}
+
+std::string ElemCorner::print() const
+{
+  std::stringstream oss;
+  if (at_vertex())
+    oss << "at vertex " << vertex();
+  else if (at_edge())
+    oss << "at edge with vertices " << first << " and " << second;
+  else
+    oss << "not at corner";
+  return oss.str();
+}
+
+bool ElemCorner::is_valid(const Elem & elem, const Point & point, const Real tol) const
+{
+  libmesh_assert(first == Elem::invalid_vertex || first < elem.n_vertices());
+  libmesh_assert(second == Elem::invalid_vertex || second < elem.n_vertices());
+
+  if (at_vertex())
+    return vertex_point(elem).relative_fuzzy_equals(point, tol);
+  if (at_edge())
+    return build_edge(elem)->contains_point(point, tol);
+
+  return true;
+}
+
+} // namespace libMesh
+
+std::ostream &
+operator<<(std::ostream & os, const libMesh::ElemCorner & elem_corner)
+{
+  os << elem_corner.print();
+  return os;
+}
diff --git a/src/geom/intersection_tools.C b/src/geom/intersection_tools.C
new file mode 100644
index 00000000000..775433ba14b
--- /dev/null
+++ b/src/geom/intersection_tools.C
@@ -0,0 +1,217 @@
+// The libMesh Finite Element Library.
+// Copyright (C) 2002-2022 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner
+
+// This library is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+// Lesser General Public License for more details.
+
+// You should have received a copy of the GNU Lesser General Public
+// License along with this library; if not, write to the Free Software
+// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+
+
+
+#include "libmesh/intersection_tools.h"
+
+#include "libmesh/point.h"
+#include "libmesh/int_range.h"
+#include "libmesh/elem.h"
+#include "libmesh/elem_corner.h"
+
+namespace libMesh::IntersectionTools
+{
+
+WithinSegmentResult within_segment(const Point & s1,
+                                   const Point & s2,
+                                   const Point & p,
+                                   const Real tol)
+{
+  libmesh_assert(!s1.relative_fuzzy_equals(s2, tol));
+
+  // First, check whether or not the points are collinear
+  const auto l = s1 - s2;
+  const auto l_norm = l.norm();
+  const auto diff1 = p - s1;
+  const auto diff1_norm = diff1.norm();
+  if ((std::abs(l * diff1) / (l_norm * diff1_norm)) < ((Real)1 - tol))
+    return NOT_WITHIN;
+
+  // If the points are collinear, make sure that p is
+  // somewhere between [s1, s2]
+  const auto diff2_norm = (p - s2).norm();
+  const auto tol_scaled = tol * l_norm;
+  if (std::abs(diff1_norm + diff2_norm - l_norm) < tol_scaled)
+  {
+    if (diff1_norm < tol_scaled)
+      return AT_BEGINNING;
+    if (diff2_norm < tol_scaled)
+      return AT_END;
+    return BETWEEN;
+  }
+
+  return NOT_WITHIN;
+}
+
+bool collinear(const Point & p1,
+               const Point & p2,
+               const Point & p3,
+               const Real tol)
+{
+  const auto p2_p1 = p1 - p2;
+  const auto p2_p1_norm = p2_p1.norm();
+  const auto p3_p2 = p2 - p3;
+  const auto p3_p2_norm = p3_p2.norm();
+  const auto denom = p2_p1_norm * p3_p2_norm;
+  if (denom == 0) // same points
+    return true;
+  return (std::abs(p2_p1 * p3_p2) / denom) > ((Real)1 - tol);
+}
+
+bool edges_are_collinear(const Elem & elem, const Real tol)
+{
+  if (elem.default_order() < 2)
+    return true;
+
+  for (const auto e : elem.edge_index_range())
+  {
+    // we should expect at least 3 nodes/edges for our higher order elems
+    libmesh_assert_greater_equal(elem.n_nodes_on_edge(e), 3);
+
+    const unsigned int * edge_nodes_map = elem.nodes_on_edge_ptr(e);
+    if (!collinear(elem.point(edge_nodes_map[0]),
+                   elem.point(edge_nodes_map[1]),
+                   elem.point(edge_nodes_map[2]),
+                   tol))
+      return false;
+  }
+
+  return true;
+}
+
+bool within_edge_on_side(const Elem & elem,
+                         const Point & p,
+                         const unsigned short s,
+                         ElemCorner & corner,
+                         const bool linearize,
+                         const Real tol)
+{
+  libmesh_assert_less(s, elem.n_sides());
+  libmesh_assert(corner.is_invalid());
+  libmesh_assert_equal_to(elem.dim(), 3);
+
+  // Without linearization, make sure that our edges are collinear
+  if (!linearize && !edges_are_collinear(elem, tol))
+    libmesh_error_msg("Failed to use Cell::without_edge_on_side without linearization "
+                      "because an edge was found that is not collinear.");
+
+  const auto vs = elem.n_vertices_on_side(s);
+  const unsigned int * side_nodes_map = elem.nodes_on_side_ptr(s);
+
+  // side_nodes_map will point to something like [v0, v1, v2, v3]
+  // With the loop below, we're going to check (in this order):
+  // [v3 -> v0], [v0 -> v1], [v1 -> v2], [v2 -> v3]
+  auto last_v = side_nodes_map[vs - 1];
+  for (const auto side_v : make_range(vs))
+  {
+    if (detail::_within_edge(elem, p, corner, last_v, side_nodes_map[side_v], tol))
+      return true;
+    last_v = side_nodes_map[side_v];
+  }
+
+  return false;
+}
+
+bool within_edge(const Elem & elem,
+                 const Point & p,
+                 ElemCorner & corner,
+                 const bool linearize,
+                 const Real tol)
+{
+  libmesh_assert(corner.is_invalid());
+
+  // Without linearization, make sure that our edges are collinear
+  if (!linearize && !edges_are_collinear(elem, tol))
+    libmesh_error_msg("Failed to use Cell::without_edge without linearization "
+                      "because an edge was found that is not collinear.");
+
+  for (const auto e : elem.edge_index_range())
+  {
+    const unsigned int * edge_nodes_map = elem.nodes_on_edge_ptr(e);
+    if (detail::_within_edge(elem, p, corner, edge_nodes_map[0], edge_nodes_map[1], tol))
+      return true;
+  }
+
+  return false;
+}
+
+unsigned short at_vertex(const Elem & elem,
+                         const Point & p,
+                         const Real tol)
+{
+  for (const auto v : elem.vertex_index_range())
+    if (elem.point(v).relative_fuzzy_equals(p, tol))
+      return v;
+  return Elem::invalid_vertex;
+}
+
+unsigned short at_vertex_on_side(const Elem & elem,
+                                 const Point & p,
+                                 const unsigned short s,
+                                 const Real tol)
+{
+  libmesh_assert_less(s, elem.n_sides());
+  const unsigned int * side_node_map = elem.nodes_on_side_ptr(s);
+  for (const auto i : make_range(elem.n_vertices_on_side(s)))
+  {
+    const auto v = side_node_map[i];
+    if (elem.point(v).relative_fuzzy_equals(p, tol))
+      return v;
+  }
+  return Elem::invalid_vertex;
+}
+
+namespace detail
+{
+bool _within_edge(const Elem & elem,
+                  const Point & p,
+                  ElemCorner & corner,
+                  const unsigned int v1,
+                  const unsigned int v2,
+                  const Real tol)
+{
+  libmesh_assert(corner.is_invalid());
+  libmesh_assert_less(v1, elem.n_vertices());
+  libmesh_assert_less(v2, elem.n_vertices());
+
+  const auto within_result = within_segment(elem.point(v1), elem.point(v2), p, tol);
+  if (within_result == NOT_WITHIN)
+    return false;
+
+  if (within_result == BETWEEN)
+  {
+    corner.set_edge(v1, v2);
+    libmesh_assert(corner.build_edge(elem)->contains_point(p, tol));
+  }
+  else if (within_result == AT_BEGINNING)
+  {
+    corner.set_vertex(v1);
+    libmesh_assert(elem.point(v1).relative_fuzzy_equals(p, tol));
+  }
+  else
+  {
+    corner.set_vertex(v2);
+    libmesh_assert(elem.point(v2).relative_fuzzy_equals(p, tol));
+  }
+
+  return true;
+}
+} // namespace detail
+} // namespace libMesh::IntersectionTools
+
+
diff --git a/src/libmesh_SOURCES b/src/libmesh_SOURCES
index 2b2b05a2b7c..6e982d1678c 100644
--- a/src/libmesh_SOURCES
+++ b/src/libmesh_SOURCES
@@ -152,6 +152,7 @@ libmesh_SOURCES =  \
         src/geom/edge_edge4.C \
         src/geom/edge_inf_edge2.C \
         src/geom/elem.C \
+        src/geom/elem_corner.C \
         src/geom/elem_cutter.C \
         src/geom/elem_quality.C \
         src/geom/elem_refinement.C \
@@ -172,6 +173,7 @@ libmesh_SOURCES =  \
         src/geom/face_tri3_subdivision.C \
         src/geom/face_tri6.C \
         src/geom/face_tri7.C \
+        src/geom/intersection_tools.C \
         src/geom/node.C \
         src/geom/node_elem.C \
         src/geom/plane.C \
diff --git a/src/mesh/xdr_io.C b/src/mesh/xdr_io.C
index 4f2e386025f..ccfb2c8b134 100644
--- a/src/mesh/xdr_io.C
+++ b/src/mesh/xdr_io.C
@@ -222,7 +222,7 @@ void XdrIO::write (const std::string & name)
 
           // Write vector of elemset ids to file with comment
           std::string comment_string = "# elemset ids for elemset code " + std::to_string(elemset_code);
-          io.data(elemset_id_vec, comment_string.c_str());
+          io.data(elemset_id_vec, comment_string);
         }
     }
 
@@ -421,7 +421,7 @@ XdrIO::write_serialized_connectivity (Xdr & io,
           legend += "p_level ";
         legend += "(n0 ... nN-1) ]";
         comment += legend;
-        io.data (n_global_elem_at_level[0], comment.c_str());
+        io.data (n_global_elem_at_level[0], comment);
       }
 
       for (auto pid : make_range(this->n_processors()))
@@ -542,7 +542,7 @@ XdrIO::write_serialized_connectivity (Xdr & io,
               buf << "p_level ";
             buf << "(n0 ... nN-1) ]";
 
-            io.data (n_global_elem_at_level[level], buf.str().c_str());
+            io.data (n_global_elem_at_level[level], buf.str());
           }
 
           for (auto pid : make_range(this->n_processors()))
@@ -1161,7 +1161,7 @@ void XdrIO::write_serialized_bcs_helper (Xdr & io, const new_header_id_type n_bc
     {
       std::stringstream comment_string;
       comment_string << "# number of " << bc_type << " boundary conditions";
-      io.data (n_bcs_out, comment_string.str().c_str());
+      io.data (n_bcs_out, comment_string.str());
     }
   n_bcs_out = 0;
 
diff --git a/src/systems/equation_systems_io.C b/src/systems/equation_systems_io.C
index ac1c82b14dd..1a4e8bac6b5 100644
--- a/src/systems/equation_systems_io.C
+++ b/src/systems/equation_systems_io.C
@@ -529,7 +529,7 @@ void EquationSystems::write(std::string_view name,
               // Note: There is no Xdr::data overload taking a "const
               // std::string &" so we need to make a copy.
               std::string copy = sys_name;
-              io.data (copy, comment.c_str());
+              io.data (copy, comment);
             }
 
             // 4.)
@@ -541,7 +541,7 @@ void EquationSystems::write(std::string_view name,
               comment =  "# Type, System No. ";
               comment += std::to_string(sys_num);
 
-              io.data (sys_type, comment.c_str());
+              io.data (sys_type, comment);
             }
 
             // 5.) - 9.)
diff --git a/src/systems/system_io.C b/src/systems/system_io.C
index 46f1dc4d19a..f78afe28655 100644
--- a/src/systems/system_io.C
+++ b/src/systems/system_io.C
@@ -1321,7 +1321,7 @@ void System::write_header (Xdr & io,
     comment += "\"";
 
     unsigned int nv = this->n_vars();
-    io.data (nv, comment.c_str());
+    io.data (nv, comment);
   }
 
 
@@ -1338,7 +1338,7 @@ void System::write_header (Xdr & io,
         comment += "\"";
 
         std::string var_name = this->variable_name(var);
-        io.data (var_name, comment.c_str());
+        io.data (var_name, comment);
       }
 
       // 6.1.) Variable subdomains
@@ -1353,7 +1353,7 @@ void System::write_header (Xdr & io,
         const std::set<subdomain_id_type> & domains = this->variable(var).active_subdomains();
         std::vector<subdomain_id_type> domain_array;
         domain_array.assign(domains.begin(), domains.end());
-        io.data (domain_array, comment.c_str());
+        io.data (domain_array, comment);
       }
 
       // 7.)
@@ -1368,7 +1368,7 @@ void System::write_header (Xdr & io,
         comment += "\"";
 
         int order = static_cast<int>(this->variable_type(var).order);
-        io.data (order, comment.c_str());
+        io.data (order, comment);
       }
 
 
@@ -1383,7 +1383,7 @@ void System::write_header (Xdr & io,
         comment += "\"";
 
         int rad_order = static_cast<int>(this->variable_type(var).radial_order);
-        io.data (rad_order, comment.c_str());
+        io.data (rad_order, comment);
       }
 
 #endif
@@ -1400,7 +1400,7 @@ void System::write_header (Xdr & io,
 
         const FEType & type = this->variable_type(var);
         int fam = static_cast<int>(type.family);
-        io.data (fam, comment.c_str());
+        io.data (fam, comment);
 
 #ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
 
@@ -1411,7 +1411,7 @@ void System::write_header (Xdr & io,
         comment += "\"";
 
         int radial_fam = static_cast<int>(type.radial_family);
-        io.data (radial_fam, comment.c_str());
+        io.data (radial_fam, comment);
 
         comment = "#     Infinite Mapping Type, Variable \"";
         comment += this->variable_name(var);
@@ -1420,7 +1420,7 @@ void System::write_header (Xdr & io,
         comment += "\"";
 
         int i_map = static_cast<int>(type.inf_map);
-        io.data (i_map, comment.c_str());
+        io.data (i_map, comment);
 #endif
       }
     } // end of the variable loop
@@ -1437,7 +1437,7 @@ void System::write_header (Xdr & io,
       comment += "\"";
 
       unsigned int nvecs = write_additional_data ? this->n_vectors () : 0;
-      io.data (nvecs, comment.c_str());
+      io.data (nvecs, comment);
     }
 
     if (write_additional_data)
@@ -1451,13 +1451,13 @@ void System::write_header (Xdr & io,
             comment =  "# Name of " + dth_vector;
             std::string vec_name = pr.first;
 
-            io.data (vec_name, comment.c_str());
+            io.data (vec_name, comment);
             int vec_projection = _vector_projections.at(vec_name);
             comment = "# Whether to do projections for " + dth_vector;
-            io.data (vec_projection, comment.c_str());
+            io.data (vec_projection, comment);
             int vec_type = _vector_types.at(vec_name);
             comment = "# Parallel type of " + dth_vector;
-            io.data (vec_type, comment.c_str());
+            io.data (vec_type, comment);
           }
       }
   }
@@ -1580,7 +1580,7 @@ void System::write_parallel_data (Xdr & io,
     comment += "\" Solution Vector";
   }
 
-  io.data (io_buffer, comment.c_str());
+  io.data (io_buffer, comment);
 
   // total_written_size += io_buffer.size();
 
@@ -1646,7 +1646,7 @@ void System::write_parallel_data (Xdr & io,
             comment += "\"";
           }
 
-          io.data (io_buffer, comment.c_str());
+          io.data (io_buffer, comment);
 
           // total_written_size += io_buffer.size();
         }
diff --git a/src/utils/xdr_cxx.C b/src/utils/xdr_cxx.C
index 49a4c2107c8..71cc010166d 100644
--- a/src/utils/xdr_cxx.C
+++ b/src/utils/xdr_cxx.C
@@ -741,7 +741,7 @@ void Xdr::do_write(std::vector<std::complex<T>> & a)
 
 
 template <typename T>
-void Xdr::data (T & a, const char * comment_in)
+void Xdr::data (T & a, std::string_view comment_in)
 {
   switch (mode)
     {
@@ -790,7 +790,7 @@ void Xdr::data (T & a, const char * comment_in)
 
         // If there's a comment provided, write a tab character and
         // then the comment.
-        if (std::string(comment_in) != "")
+        if (comment_in != "")
           *out << "\t " << comment_in;
 
         // Go to the next line.
@@ -1297,42 +1297,42 @@ void Xdr::comment (std::string & comment_in)
 
 
 //
-template LIBMESH_EXPORT void Xdr::data<int>                              (int &,                             const char *);
-template LIBMESH_EXPORT void Xdr::data<unsigned int>                     (unsigned int &,                    const char *);
-template LIBMESH_EXPORT void Xdr::data<unsigned short int>               (unsigned short int &,              const char *);
-template LIBMESH_EXPORT void Xdr::data<short int>                        (short int &,                       const char *);
-template LIBMESH_EXPORT void Xdr::data<unsigned long int>                (unsigned long int &,               const char *);
-template LIBMESH_EXPORT void Xdr::data<unsigned long long>               (unsigned long long &,              const char *);
-template LIBMESH_EXPORT void Xdr::data<long int>                         (long int &,                        const char *);
-template LIBMESH_EXPORT void Xdr::data<long long>                        (long long &,                       const char *);
-template LIBMESH_EXPORT void Xdr::data<char>                             (char &,                            const char *);
-template LIBMESH_EXPORT void Xdr::data<signed char>                      (signed char &,                     const char *);
-template LIBMESH_EXPORT void Xdr::data<unsigned char>                    (unsigned char &,                   const char *);
-template LIBMESH_EXPORT void Xdr::data<float>                            (float &,                           const char *);
-template LIBMESH_EXPORT void Xdr::data<double>                           (double &,                          const char *);
-template LIBMESH_EXPORT void Xdr::data<long double>                      (long double &,                     const char *);
-template LIBMESH_EXPORT void Xdr::data<std::complex<float>>              (std::complex<float> &,             const char *);
-template LIBMESH_EXPORT void Xdr::data<std::complex<double>>             (std::complex<double> &,            const char *);
-template LIBMESH_EXPORT void Xdr::data<std::complex<long double>>        (std::complex<long double> &,       const char *);
-template LIBMESH_EXPORT void Xdr::data<std::string>                      (std::string &,                     const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<int>>                 (std::vector<int> &,                const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned int>>        (std::vector<unsigned int> &,       const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<short int>>           (std::vector<short int> &,          const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned short int>>  (std::vector<unsigned short int> &, const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<long int>>            (std::vector<long int> &,           const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<long long>>           (std::vector<long long> &,          const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned long int>>   (std::vector<unsigned long int> &,  const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned long long>>  (std::vector<unsigned long long> &, const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<char>>                (std::vector<char> &,               const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<signed char>>         (std::vector<signed char> &,        const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned char>>       (std::vector<unsigned char> &,      const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<float>>               (std::vector<float> &,              const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<double>>              (std::vector<double> &,             const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<long double>>         (std::vector<long double> &,        const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<std::complex<float>>>  (std::vector<std::complex<float>> &,  const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<std::complex<double>>> (std::vector<std::complex<double>> &, const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<std::complex<long double>>> (std::vector<std::complex<long double>> &, const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<std::string>>        (std::vector<std::string> &,        const char *);
+template LIBMESH_EXPORT void Xdr::data<int>                              (int &,                             std::string_view);
+template LIBMESH_EXPORT void Xdr::data<unsigned int>                     (unsigned int &,                    std::string_view);
+template LIBMESH_EXPORT void Xdr::data<unsigned short int>               (unsigned short int &,              std::string_view);
+template LIBMESH_EXPORT void Xdr::data<short int>                        (short int &,                       std::string_view);
+template LIBMESH_EXPORT void Xdr::data<unsigned long int>                (unsigned long int &,               std::string_view);
+template LIBMESH_EXPORT void Xdr::data<unsigned long long>               (unsigned long long &,              std::string_view);
+template LIBMESH_EXPORT void Xdr::data<long int>                         (long int &,                        std::string_view);
+template LIBMESH_EXPORT void Xdr::data<long long>                        (long long &,                       std::string_view);
+template LIBMESH_EXPORT void Xdr::data<char>                             (char &,                            std::string_view);
+template LIBMESH_EXPORT void Xdr::data<signed char>                      (signed char &,                     std::string_view);
+template LIBMESH_EXPORT void Xdr::data<unsigned char>                    (unsigned char &,                   std::string_view);
+template LIBMESH_EXPORT void Xdr::data<float>                            (float &,                           std::string_view);
+template LIBMESH_EXPORT void Xdr::data<double>                           (double &,                          std::string_view);
+template LIBMESH_EXPORT void Xdr::data<long double>                      (long double &,                     std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::complex<float>>              (std::complex<float> &,             std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::complex<double>>             (std::complex<double> &,            std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::complex<long double>>        (std::complex<long double> &,       std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::string>                      (std::string &,                     std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<int>>                 (std::vector<int> &,                std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned int>>        (std::vector<unsigned int> &,       std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<short int>>           (std::vector<short int> &,          std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned short int>>  (std::vector<unsigned short int> &, std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<long int>>            (std::vector<long int> &,           std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<long long>>           (std::vector<long long> &,          std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned long int>>   (std::vector<unsigned long int> &,  std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned long long>>  (std::vector<unsigned long long> &, std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<char>>                (std::vector<char> &,               std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<signed char>>         (std::vector<signed char> &,        std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<unsigned char>>       (std::vector<unsigned char> &,      std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<float>>               (std::vector<float> &,              std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<double>>              (std::vector<double> &,             std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<long double>>         (std::vector<long double> &,        std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<std::complex<float>>>  (std::vector<std::complex<float>> &,  std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<std::complex<double>>> (std::vector<std::complex<double>> &, std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<std::complex<long double>>> (std::vector<std::complex<long double>> &, std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<std::string>>        (std::vector<std::string> &,        std::string_view);
 template LIBMESH_EXPORT void Xdr::data_stream<unsigned char>      (unsigned char * val,      const unsigned int len, const unsigned int line_break);
 template LIBMESH_EXPORT void Xdr::data_stream<short int>          (short int * val,          const unsigned int len, const unsigned int line_break);
 template LIBMESH_EXPORT void Xdr::data_stream<int>                (int * val,                const unsigned int len, const unsigned int line_break);
@@ -1343,10 +1343,10 @@ template LIBMESH_EXPORT void Xdr::data_stream<unsigned long int>  (unsigned long
 template LIBMESH_EXPORT void Xdr::data_stream<unsigned long long> (unsigned long long * val, const unsigned int len, const unsigned int line_break);
 
 #ifdef LIBMESH_DEFAULT_QUADRUPLE_PRECISION
-template LIBMESH_EXPORT void Xdr::data<Real>                             (Real &,                            const char *);
-template LIBMESH_EXPORT void Xdr::data<std::complex<Real>>               (std::complex<Real> &,              const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<Real>>                (std::vector<Real> &,               const char *);
-template LIBMESH_EXPORT void Xdr::data<std::vector<std::complex<Real>>>  (std::vector<std::complex<Real>> &, const char *);
+template LIBMESH_EXPORT void Xdr::data<Real>                             (Real &,                            std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::complex<Real>>               (std::complex<Real> &,              std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<Real>>                (std::vector<Real> &,               std::string_view);
+template LIBMESH_EXPORT void Xdr::data<std::vector<std::complex<Real>>>  (std::vector<std::complex<Real>> &, std::string_view);
 #endif
 
 } // namespace libMesh
diff --git a/tests/Makefile.am b/tests/Makefile.am
index e8e8e7d1a0c..8cc0290a91e 100644
--- a/tests/Makefile.am
+++ b/tests/Makefile.am
@@ -45,7 +45,9 @@ unit_tests_sources = \
   fe/dual_shape_verification_test.C \
   geom/bbox_test.C \
   geom/edge_test.C \
+  geom/elem_corner_test.C \
   geom/elem_test.C \
+  geom/intersection_tools_test.C \
   geom/node_test.C \
   geom/point_test.C \
   geom/point_test.h \
diff --git a/tests/Makefile.in b/tests/Makefile.in
index f298e558e57..b42e29d947f 100644
--- a/tests/Makefile.in
+++ b/tests/Makefile.in
@@ -206,7 +206,8 @@ am__unit_tests_dbg_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	fe/fe_rational_map.C fe/fe_rational_test.C fe/fe_side_test.C \
 	fe/fe_szabab_test.C fe/fe_test.h fe/fe_xyz_test.C \
 	fe/dual_shape_verification_test.C geom/bbox_test.C \
-	geom/edge_test.C geom/elem_test.C geom/node_test.C \
+	geom/edge_test.C geom/elem_corner_test.C geom/elem_test.C \
+	geom/intersection_tools_test.C geom/node_test.C \
 	geom/point_test.C geom/point_test.h geom/side_test.C \
 	geom/volume_test.C geom/which_node_am_i_test.C \
 	mesh/all_second_order.C mesh/all_tri.C mesh/distort.C \
@@ -293,7 +294,9 @@ am__objects_3 = unit_tests_dbg-driver.$(OBJEXT) \
 	fe/unit_tests_dbg-dual_shape_verification_test.$(OBJEXT) \
 	geom/unit_tests_dbg-bbox_test.$(OBJEXT) \
 	geom/unit_tests_dbg-edge_test.$(OBJEXT) \
+	geom/unit_tests_dbg-elem_corner_test.$(OBJEXT) \
 	geom/unit_tests_dbg-elem_test.$(OBJEXT) \
+	geom/unit_tests_dbg-intersection_tools_test.$(OBJEXT) \
 	geom/unit_tests_dbg-node_test.$(OBJEXT) \
 	geom/unit_tests_dbg-point_test.$(OBJEXT) \
 	geom/unit_tests_dbg-side_test.$(OBJEXT) \
@@ -395,7 +398,8 @@ am__unit_tests_devel_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	fe/fe_rational_map.C fe/fe_rational_test.C fe/fe_side_test.C \
 	fe/fe_szabab_test.C fe/fe_test.h fe/fe_xyz_test.C \
 	fe/dual_shape_verification_test.C geom/bbox_test.C \
-	geom/edge_test.C geom/elem_test.C geom/node_test.C \
+	geom/edge_test.C geom/elem_corner_test.C geom/elem_test.C \
+	geom/intersection_tools_test.C geom/node_test.C \
 	geom/point_test.C geom/point_test.h geom/side_test.C \
 	geom/volume_test.C geom/which_node_am_i_test.C \
 	mesh/all_second_order.C mesh/all_tri.C mesh/distort.C \
@@ -480,7 +484,9 @@ am__objects_5 = unit_tests_devel-driver.$(OBJEXT) \
 	fe/unit_tests_devel-dual_shape_verification_test.$(OBJEXT) \
 	geom/unit_tests_devel-bbox_test.$(OBJEXT) \
 	geom/unit_tests_devel-edge_test.$(OBJEXT) \
+	geom/unit_tests_devel-elem_corner_test.$(OBJEXT) \
 	geom/unit_tests_devel-elem_test.$(OBJEXT) \
+	geom/unit_tests_devel-intersection_tools_test.$(OBJEXT) \
 	geom/unit_tests_devel-node_test.$(OBJEXT) \
 	geom/unit_tests_devel-point_test.$(OBJEXT) \
 	geom/unit_tests_devel-side_test.$(OBJEXT) \
@@ -578,7 +584,8 @@ am__unit_tests_oprof_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	fe/fe_rational_map.C fe/fe_rational_test.C fe/fe_side_test.C \
 	fe/fe_szabab_test.C fe/fe_test.h fe/fe_xyz_test.C \
 	fe/dual_shape_verification_test.C geom/bbox_test.C \
-	geom/edge_test.C geom/elem_test.C geom/node_test.C \
+	geom/edge_test.C geom/elem_corner_test.C geom/elem_test.C \
+	geom/intersection_tools_test.C geom/node_test.C \
 	geom/point_test.C geom/point_test.h geom/side_test.C \
 	geom/volume_test.C geom/which_node_am_i_test.C \
 	mesh/all_second_order.C mesh/all_tri.C mesh/distort.C \
@@ -663,7 +670,9 @@ am__objects_7 = unit_tests_oprof-driver.$(OBJEXT) \
 	fe/unit_tests_oprof-dual_shape_verification_test.$(OBJEXT) \
 	geom/unit_tests_oprof-bbox_test.$(OBJEXT) \
 	geom/unit_tests_oprof-edge_test.$(OBJEXT) \
+	geom/unit_tests_oprof-elem_corner_test.$(OBJEXT) \
 	geom/unit_tests_oprof-elem_test.$(OBJEXT) \
+	geom/unit_tests_oprof-intersection_tools_test.$(OBJEXT) \
 	geom/unit_tests_oprof-node_test.$(OBJEXT) \
 	geom/unit_tests_oprof-point_test.$(OBJEXT) \
 	geom/unit_tests_oprof-side_test.$(OBJEXT) \
@@ -761,7 +770,8 @@ am__unit_tests_opt_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	fe/fe_rational_map.C fe/fe_rational_test.C fe/fe_side_test.C \
 	fe/fe_szabab_test.C fe/fe_test.h fe/fe_xyz_test.C \
 	fe/dual_shape_verification_test.C geom/bbox_test.C \
-	geom/edge_test.C geom/elem_test.C geom/node_test.C \
+	geom/edge_test.C geom/elem_corner_test.C geom/elem_test.C \
+	geom/intersection_tools_test.C geom/node_test.C \
 	geom/point_test.C geom/point_test.h geom/side_test.C \
 	geom/volume_test.C geom/which_node_am_i_test.C \
 	mesh/all_second_order.C mesh/all_tri.C mesh/distort.C \
@@ -846,7 +856,9 @@ am__objects_9 = unit_tests_opt-driver.$(OBJEXT) \
 	fe/unit_tests_opt-dual_shape_verification_test.$(OBJEXT) \
 	geom/unit_tests_opt-bbox_test.$(OBJEXT) \
 	geom/unit_tests_opt-edge_test.$(OBJEXT) \
+	geom/unit_tests_opt-elem_corner_test.$(OBJEXT) \
 	geom/unit_tests_opt-elem_test.$(OBJEXT) \
+	geom/unit_tests_opt-intersection_tools_test.$(OBJEXT) \
 	geom/unit_tests_opt-node_test.$(OBJEXT) \
 	geom/unit_tests_opt-point_test.$(OBJEXT) \
 	geom/unit_tests_opt-side_test.$(OBJEXT) \
@@ -944,7 +956,8 @@ am__unit_tests_prof_SOURCES_DIST = driver.C libmesh_cppunit.h \
 	fe/fe_rational_map.C fe/fe_rational_test.C fe/fe_side_test.C \
 	fe/fe_szabab_test.C fe/fe_test.h fe/fe_xyz_test.C \
 	fe/dual_shape_verification_test.C geom/bbox_test.C \
-	geom/edge_test.C geom/elem_test.C geom/node_test.C \
+	geom/edge_test.C geom/elem_corner_test.C geom/elem_test.C \
+	geom/intersection_tools_test.C geom/node_test.C \
 	geom/point_test.C geom/point_test.h geom/side_test.C \
 	geom/volume_test.C geom/which_node_am_i_test.C \
 	mesh/all_second_order.C mesh/all_tri.C mesh/distort.C \
@@ -1029,7 +1042,9 @@ am__objects_11 = unit_tests_prof-driver.$(OBJEXT) \
 	fe/unit_tests_prof-dual_shape_verification_test.$(OBJEXT) \
 	geom/unit_tests_prof-bbox_test.$(OBJEXT) \
 	geom/unit_tests_prof-edge_test.$(OBJEXT) \
+	geom/unit_tests_prof-elem_corner_test.$(OBJEXT) \
 	geom/unit_tests_prof-elem_test.$(OBJEXT) \
+	geom/unit_tests_prof-intersection_tools_test.$(OBJEXT) \
 	geom/unit_tests_prof-node_test.$(OBJEXT) \
 	geom/unit_tests_prof-point_test.$(OBJEXT) \
 	geom/unit_tests_prof-side_test.$(OBJEXT) \
@@ -1246,7 +1261,9 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	fparser/$(DEPDIR)/unit_tests_prof-autodiff.Po \
 	geom/$(DEPDIR)/unit_tests_dbg-bbox_test.Po \
 	geom/$(DEPDIR)/unit_tests_dbg-edge_test.Po \
+	geom/$(DEPDIR)/unit_tests_dbg-elem_corner_test.Po \
 	geom/$(DEPDIR)/unit_tests_dbg-elem_test.Po \
+	geom/$(DEPDIR)/unit_tests_dbg-intersection_tools_test.Po \
 	geom/$(DEPDIR)/unit_tests_dbg-node_test.Po \
 	geom/$(DEPDIR)/unit_tests_dbg-point_test.Po \
 	geom/$(DEPDIR)/unit_tests_dbg-side_test.Po \
@@ -1254,7 +1271,9 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	geom/$(DEPDIR)/unit_tests_dbg-which_node_am_i_test.Po \
 	geom/$(DEPDIR)/unit_tests_devel-bbox_test.Po \
 	geom/$(DEPDIR)/unit_tests_devel-edge_test.Po \
+	geom/$(DEPDIR)/unit_tests_devel-elem_corner_test.Po \
 	geom/$(DEPDIR)/unit_tests_devel-elem_test.Po \
+	geom/$(DEPDIR)/unit_tests_devel-intersection_tools_test.Po \
 	geom/$(DEPDIR)/unit_tests_devel-node_test.Po \
 	geom/$(DEPDIR)/unit_tests_devel-point_test.Po \
 	geom/$(DEPDIR)/unit_tests_devel-side_test.Po \
@@ -1262,7 +1281,9 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	geom/$(DEPDIR)/unit_tests_devel-which_node_am_i_test.Po \
 	geom/$(DEPDIR)/unit_tests_oprof-bbox_test.Po \
 	geom/$(DEPDIR)/unit_tests_oprof-edge_test.Po \
+	geom/$(DEPDIR)/unit_tests_oprof-elem_corner_test.Po \
 	geom/$(DEPDIR)/unit_tests_oprof-elem_test.Po \
+	geom/$(DEPDIR)/unit_tests_oprof-intersection_tools_test.Po \
 	geom/$(DEPDIR)/unit_tests_oprof-node_test.Po \
 	geom/$(DEPDIR)/unit_tests_oprof-point_test.Po \
 	geom/$(DEPDIR)/unit_tests_oprof-side_test.Po \
@@ -1270,7 +1291,9 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	geom/$(DEPDIR)/unit_tests_oprof-which_node_am_i_test.Po \
 	geom/$(DEPDIR)/unit_tests_opt-bbox_test.Po \
 	geom/$(DEPDIR)/unit_tests_opt-edge_test.Po \
+	geom/$(DEPDIR)/unit_tests_opt-elem_corner_test.Po \
 	geom/$(DEPDIR)/unit_tests_opt-elem_test.Po \
+	geom/$(DEPDIR)/unit_tests_opt-intersection_tools_test.Po \
 	geom/$(DEPDIR)/unit_tests_opt-node_test.Po \
 	geom/$(DEPDIR)/unit_tests_opt-point_test.Po \
 	geom/$(DEPDIR)/unit_tests_opt-side_test.Po \
@@ -1278,7 +1301,9 @@ am__depfiles_remade = ./$(DEPDIR)/unit_tests_dbg-driver.Po \
 	geom/$(DEPDIR)/unit_tests_opt-which_node_am_i_test.Po \
 	geom/$(DEPDIR)/unit_tests_prof-bbox_test.Po \
 	geom/$(DEPDIR)/unit_tests_prof-edge_test.Po \
+	geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Po \
 	geom/$(DEPDIR)/unit_tests_prof-elem_test.Po \
+	geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Po \
 	geom/$(DEPDIR)/unit_tests_prof-node_test.Po \
 	geom/$(DEPDIR)/unit_tests_prof-point_test.Po \
 	geom/$(DEPDIR)/unit_tests_prof-side_test.Po \
@@ -2106,7 +2131,8 @@ unit_tests_sources = driver.C libmesh_cppunit.h stream_redirector.h \
 	fe/fe_rational_map.C fe/fe_rational_test.C fe/fe_side_test.C \
 	fe/fe_szabab_test.C fe/fe_test.h fe/fe_xyz_test.C \
 	fe/dual_shape_verification_test.C geom/bbox_test.C \
-	geom/edge_test.C geom/elem_test.C geom/node_test.C \
+	geom/edge_test.C geom/elem_corner_test.C geom/elem_test.C \
+	geom/intersection_tools_test.C geom/node_test.C \
 	geom/point_test.C geom/point_test.h geom/side_test.C \
 	geom/volume_test.C geom/which_node_am_i_test.C \
 	mesh/all_second_order.C mesh/all_tri.C mesh/distort.C \
@@ -2339,8 +2365,12 @@ geom/unit_tests_dbg-bbox_test.$(OBJEXT): geom/$(am__dirstamp) \
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+
 geom/unit_tests_devel-elem_test.o: geom/elem_test.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_devel_CPPFLAGS) $(CPPFLAGS) $(unit_tests_devel_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_devel-elem_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_devel-elem_test.Tpo -c -o geom/unit_tests_devel-elem_test.o `test -f 'geom/elem_test.C' || echo '$(srcdir)/'`geom/elem_test.C
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@@ -5755,6 +5853,20 @@ geom/unit_tests_devel-elem_test.obj: geom/elem_test.C
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+geom/unit_tests_devel-intersection_tools_test.o: geom/intersection_tools_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_devel_CPPFLAGS) $(CPPFLAGS) $(unit_tests_devel_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_devel-intersection_tools_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_devel-intersection_tools_test.Tpo -c -o geom/unit_tests_devel-intersection_tools_test.o `test -f 'geom/intersection_tools_test.C' || echo '$(srcdir)/'`geom/intersection_tools_test.C
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+
+geom/unit_tests_devel-intersection_tools_test.obj: geom/intersection_tools_test.C
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+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/intersection_tools_test.C' object='geom/unit_tests_devel-intersection_tools_test.obj' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
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+
 geom/unit_tests_devel-node_test.o: geom/node_test.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_devel_CPPFLAGS) $(CPPFLAGS) $(unit_tests_devel_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_devel-node_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_devel-node_test.Tpo -c -o geom/unit_tests_devel-node_test.o `test -f 'geom/node_test.C' || echo '$(srcdir)/'`geom/node_test.C
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@@ -7169,6 +7281,20 @@ geom/unit_tests_oprof-edge_test.obj: geom/edge_test.C
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+geom/unit_tests_oprof-elem_corner_test.o: geom/elem_corner_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_oprof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_oprof_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_oprof-elem_corner_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_oprof-elem_corner_test.Tpo -c -o geom/unit_tests_oprof-elem_corner_test.o `test -f 'geom/elem_corner_test.C' || echo '$(srcdir)/'`geom/elem_corner_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_oprof-elem_corner_test.Tpo geom/$(DEPDIR)/unit_tests_oprof-elem_corner_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/elem_corner_test.C' object='geom/unit_tests_oprof-elem_corner_test.o' libtool=no @AMDEPBACKSLASH@
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+
+geom/unit_tests_oprof-elem_corner_test.obj: geom/elem_corner_test.C
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+
 geom/unit_tests_oprof-elem_test.o: geom/elem_test.C
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@@ -7183,6 +7309,20 @@ geom/unit_tests_oprof-elem_test.obj: geom/elem_test.C
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+geom/unit_tests_oprof-intersection_tools_test.o: geom/intersection_tools_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_oprof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_oprof_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_oprof-intersection_tools_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_oprof-intersection_tools_test.Tpo -c -o geom/unit_tests_oprof-intersection_tools_test.o `test -f 'geom/intersection_tools_test.C' || echo '$(srcdir)/'`geom/intersection_tools_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_oprof-intersection_tools_test.Tpo geom/$(DEPDIR)/unit_tests_oprof-intersection_tools_test.Po
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+
+geom/unit_tests_oprof-intersection_tools_test.obj: geom/intersection_tools_test.C
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+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/intersection_tools_test.C' object='geom/unit_tests_oprof-intersection_tools_test.obj' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
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+
 geom/unit_tests_oprof-node_test.o: geom/node_test.C
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@@ -8597,6 +8737,20 @@ geom/unit_tests_opt-edge_test.obj: geom/edge_test.C
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+geom/unit_tests_opt-elem_corner_test.o: geom/elem_corner_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_opt-elem_corner_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_opt-elem_corner_test.Tpo -c -o geom/unit_tests_opt-elem_corner_test.o `test -f 'geom/elem_corner_test.C' || echo '$(srcdir)/'`geom/elem_corner_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_opt-elem_corner_test.Tpo geom/$(DEPDIR)/unit_tests_opt-elem_corner_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/elem_corner_test.C' object='geom/unit_tests_opt-elem_corner_test.o' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
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+
+geom/unit_tests_opt-elem_corner_test.obj: geom/elem_corner_test.C
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+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/elem_corner_test.C' object='geom/unit_tests_opt-elem_corner_test.obj' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
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+
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@@ -8611,6 +8765,20 @@ geom/unit_tests_opt-elem_test.obj: geom/elem_test.C
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+geom/unit_tests_opt-intersection_tools_test.o: geom/intersection_tools_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_opt-intersection_tools_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_opt-intersection_tools_test.Tpo -c -o geom/unit_tests_opt-intersection_tools_test.o `test -f 'geom/intersection_tools_test.C' || echo '$(srcdir)/'`geom/intersection_tools_test.C
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+
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+
 geom/unit_tests_opt-node_test.o: geom/node_test.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_opt_CPPFLAGS) $(CPPFLAGS) $(unit_tests_opt_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_opt-node_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_opt-node_test.Tpo -c -o geom/unit_tests_opt-node_test.o `test -f 'geom/node_test.C' || echo '$(srcdir)/'`geom/node_test.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_opt-node_test.Tpo geom/$(DEPDIR)/unit_tests_opt-node_test.Po
@@ -10025,6 +10193,20 @@ geom/unit_tests_prof-edge_test.obj: geom/edge_test.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o geom/unit_tests_prof-edge_test.obj `if test -f 'geom/edge_test.C'; then $(CYGPATH_W) 'geom/edge_test.C'; else $(CYGPATH_W) '$(srcdir)/geom/edge_test.C'; fi`
 
+geom/unit_tests_prof-elem_corner_test.o: geom/elem_corner_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_prof-elem_corner_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Tpo -c -o geom/unit_tests_prof-elem_corner_test.o `test -f 'geom/elem_corner_test.C' || echo '$(srcdir)/'`geom/elem_corner_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Tpo geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/elem_corner_test.C' object='geom/unit_tests_prof-elem_corner_test.o' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o geom/unit_tests_prof-elem_corner_test.o `test -f 'geom/elem_corner_test.C' || echo '$(srcdir)/'`geom/elem_corner_test.C
+
+geom/unit_tests_prof-elem_corner_test.obj: geom/elem_corner_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_prof-elem_corner_test.obj -MD -MP -MF geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Tpo -c -o geom/unit_tests_prof-elem_corner_test.obj `if test -f 'geom/elem_corner_test.C'; then $(CYGPATH_W) 'geom/elem_corner_test.C'; else $(CYGPATH_W) '$(srcdir)/geom/elem_corner_test.C'; fi`
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Tpo geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/elem_corner_test.C' object='geom/unit_tests_prof-elem_corner_test.obj' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o geom/unit_tests_prof-elem_corner_test.obj `if test -f 'geom/elem_corner_test.C'; then $(CYGPATH_W) 'geom/elem_corner_test.C'; else $(CYGPATH_W) '$(srcdir)/geom/elem_corner_test.C'; fi`
+
 geom/unit_tests_prof-elem_test.o: geom/elem_test.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_prof-elem_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_prof-elem_test.Tpo -c -o geom/unit_tests_prof-elem_test.o `test -f 'geom/elem_test.C' || echo '$(srcdir)/'`geom/elem_test.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_prof-elem_test.Tpo geom/$(DEPDIR)/unit_tests_prof-elem_test.Po
@@ -10039,6 +10221,20 @@ geom/unit_tests_prof-elem_test.obj: geom/elem_test.C
 @AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
 @am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o geom/unit_tests_prof-elem_test.obj `if test -f 'geom/elem_test.C'; then $(CYGPATH_W) 'geom/elem_test.C'; else $(CYGPATH_W) '$(srcdir)/geom/elem_test.C'; fi`
 
+geom/unit_tests_prof-intersection_tools_test.o: geom/intersection_tools_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_prof-intersection_tools_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Tpo -c -o geom/unit_tests_prof-intersection_tools_test.o `test -f 'geom/intersection_tools_test.C' || echo '$(srcdir)/'`geom/intersection_tools_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Tpo geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/intersection_tools_test.C' object='geom/unit_tests_prof-intersection_tools_test.o' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o geom/unit_tests_prof-intersection_tools_test.o `test -f 'geom/intersection_tools_test.C' || echo '$(srcdir)/'`geom/intersection_tools_test.C
+
+geom/unit_tests_prof-intersection_tools_test.obj: geom/intersection_tools_test.C
+@am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_prof-intersection_tools_test.obj -MD -MP -MF geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Tpo -c -o geom/unit_tests_prof-intersection_tools_test.obj `if test -f 'geom/intersection_tools_test.C'; then $(CYGPATH_W) 'geom/intersection_tools_test.C'; else $(CYGPATH_W) '$(srcdir)/geom/intersection_tools_test.C'; fi`
+@am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Tpo geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Po
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	$(AM_V_CXX)source='geom/intersection_tools_test.C' object='geom/unit_tests_prof-intersection_tools_test.obj' libtool=no @AMDEPBACKSLASH@
+@AMDEP_TRUE@@am__fastdepCXX_FALSE@	DEPDIR=$(DEPDIR) $(CXXDEPMODE) $(depcomp) @AMDEPBACKSLASH@
+@am__fastdepCXX_FALSE@	$(AM_V_CXX@am__nodep@)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -c -o geom/unit_tests_prof-intersection_tools_test.obj `if test -f 'geom/intersection_tools_test.C'; then $(CYGPATH_W) 'geom/intersection_tools_test.C'; else $(CYGPATH_W) '$(srcdir)/geom/intersection_tools_test.C'; fi`
+
 geom/unit_tests_prof-node_test.o: geom/node_test.C
 @am__fastdepCXX_TRUE@	$(AM_V_CXX)$(CXX) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(unit_tests_prof_CPPFLAGS) $(CPPFLAGS) $(unit_tests_prof_CXXFLAGS) $(CXXFLAGS) -MT geom/unit_tests_prof-node_test.o -MD -MP -MF geom/$(DEPDIR)/unit_tests_prof-node_test.Tpo -c -o geom/unit_tests_prof-node_test.o `test -f 'geom/node_test.C' || echo '$(srcdir)/'`geom/node_test.C
 @am__fastdepCXX_TRUE@	$(AM_V_at)$(am__mv) geom/$(DEPDIR)/unit_tests_prof-node_test.Tpo geom/$(DEPDIR)/unit_tests_prof-node_test.Po
@@ -11596,7 +11792,9 @@ distclean: distclean-am
 	-rm -f fparser/$(DEPDIR)/unit_tests_prof-autodiff.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_dbg-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_dbg-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-side_test.Po
@@ -11604,7 +11802,9 @@ distclean: distclean-am
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-which_node_am_i_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_devel-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_devel-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-side_test.Po
@@ -11612,7 +11812,9 @@ distclean: distclean-am
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-which_node_am_i_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_oprof-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_oprof-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-side_test.Po
@@ -11620,7 +11822,9 @@ distclean: distclean-am
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-which_node_am_i_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_opt-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_opt-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-side_test.Po
@@ -11628,7 +11832,9 @@ distclean: distclean-am
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-which_node_am_i_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-side_test.Po
@@ -12153,7 +12359,9 @@ maintainer-clean: maintainer-clean-am
 	-rm -f fparser/$(DEPDIR)/unit_tests_prof-autodiff.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_dbg-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_dbg-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-side_test.Po
@@ -12161,7 +12369,9 @@ maintainer-clean: maintainer-clean-am
 	-rm -f geom/$(DEPDIR)/unit_tests_dbg-which_node_am_i_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_devel-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_devel-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-side_test.Po
@@ -12169,7 +12379,9 @@ maintainer-clean: maintainer-clean-am
 	-rm -f geom/$(DEPDIR)/unit_tests_devel-which_node_am_i_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_oprof-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_oprof-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-side_test.Po
@@ -12177,7 +12389,9 @@ maintainer-clean: maintainer-clean-am
 	-rm -f geom/$(DEPDIR)/unit_tests_oprof-which_node_am_i_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_opt-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_opt-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-side_test.Po
@@ -12185,7 +12399,9 @@ maintainer-clean: maintainer-clean-am
 	-rm -f geom/$(DEPDIR)/unit_tests_opt-which_node_am_i_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-bbox_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-edge_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_prof-elem_corner_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-elem_test.Po
+	-rm -f geom/$(DEPDIR)/unit_tests_prof-intersection_tools_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-node_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-point_test.Po
 	-rm -f geom/$(DEPDIR)/unit_tests_prof-side_test.Po
diff --git a/tests/geom/elem_corner_test.C b/tests/geom/elem_corner_test.C
new file mode 100644
index 00000000000..c8e5fc917f9
--- /dev/null
+++ b/tests/geom/elem_corner_test.C
@@ -0,0 +1,177 @@
+#include <libmesh/elem_corner.h>
+#include <libmesh/mesh.h>
+#include <libmesh/mesh_generation.h>
+
+#include "test_comm.h"
+
+#include "libmesh_cppunit.h"
+
+using namespace libMesh;
+
+#if LIBMESH_DIM > 1
+class ElemCornerTest : public CppUnit::TestCase
+{
+
+public:
+  LIBMESH_CPPUNIT_TEST_SUITE( ElemCornerTest );
+  CPPUNIT_TEST( test );
+  CPPUNIT_TEST_SUITE_END();
+
+public:
+
+  void test()
+  {
+    LOG_UNIT_TEST;
+
+    // Default constructor: unvalid state
+    const ElemCorner default_corner;
+    CPPUNIT_ASSERT(default_corner.is_invalid());
+    CPPUNIT_ASSERT(!default_corner.at_corner());
+    CPPUNIT_ASSERT(!default_corner.at_vertex());
+    CPPUNIT_ASSERT(!default_corner.at_edge());
+    const std::string default_corner_string = "not at corner";
+    std::stringstream default_corner_ss;
+    default_corner_ss << default_corner;
+    CPPUNIT_ASSERT_EQUAL(default_corner_string, default_corner.print());
+    CPPUNIT_ASSERT_EQUAL(default_corner_string, default_corner_ss.str());
+
+    // EDGE elem for vertex testing
+    Mesh mesh_1d (*TestCommWorld);
+    MeshTools::Generation::build_line(mesh_1d, 1);
+    const auto edge = mesh_1d.elem_ptr(0);
+
+    // At edge vertex
+    const ElemCorner at_edge_vertex(1, Elem::invalid_vertex);
+    CPPUNIT_ASSERT(!at_edge_vertex.is_invalid());
+    CPPUNIT_ASSERT(at_edge_vertex.at_corner());
+    CPPUNIT_ASSERT(at_edge_vertex.at_vertex());
+    CPPUNIT_ASSERT(!at_edge_vertex.at_edge());
+    CPPUNIT_ASSERT(at_edge_vertex.at_vertex(1));
+    CPPUNIT_ASSERT_EQUAL(at_edge_vertex.vertex(), (unsigned short)1);
+    if (edge)
+      for (const auto v : edge->vertex_index_range())
+      {
+        CPPUNIT_ASSERT_EQUAL(at_edge_vertex.at_vertex(v), v == 1);
+        CPPUNIT_ASSERT_EQUAL(at_edge_vertex.is_valid(*edge, edge->point(v)), v == 1);
+      }
+    const std::string at_vertex_string = "at vertex 1";
+    std::stringstream at_vertex_ss;
+    at_vertex_ss << at_edge_vertex;
+    CPPUNIT_ASSERT_EQUAL(at_vertex_string, at_edge_vertex.print());
+    CPPUNIT_ASSERT_EQUAL(at_vertex_string, at_vertex_ss.str());
+
+#if LIBMESH_DIM > 1
+    // QUAD elem for vertex testing
+    Mesh mesh_2d (*TestCommWorld);
+    MeshTools::Generation::build_square (mesh_2d, 1, 1);
+    const auto quad = mesh_2d.elem_ptr(0);
+
+    // At quad vertex
+    const ElemCorner at_quad_vertex(0, Elem::invalid_vertex);
+    CPPUNIT_ASSERT(!at_quad_vertex.is_invalid());
+    CPPUNIT_ASSERT(at_quad_vertex.at_corner());
+    CPPUNIT_ASSERT(at_quad_vertex.at_vertex());
+    CPPUNIT_ASSERT(!at_quad_vertex.at_edge());
+    CPPUNIT_ASSERT(!at_quad_vertex.at_edge(0, 1));
+    CPPUNIT_ASSERT(at_quad_vertex.at_vertex(0));
+    CPPUNIT_ASSERT(!at_quad_vertex.at_vertex(1));
+    CPPUNIT_ASSERT_EQUAL(at_quad_vertex.vertex(), (unsigned short)0);
+    if (quad)
+    {
+      for (const auto v : quad->vertex_index_range())
+        CPPUNIT_ASSERT(default_corner.is_valid(*quad, quad->point(v)));
+      for (const auto e : quad->edge_index_range())
+        CPPUNIT_ASSERT(!at_quad_vertex.at_edge(*quad, e));
+      for (const auto v : quad->vertex_index_range())
+        CPPUNIT_ASSERT_EQUAL(at_quad_vertex.is_valid(*quad, quad->point(v)), v == 0);
+    }
+
+    // At quad "edge" (a side)
+    const ElemCorner at_quad_edge(1, 2);
+    CPPUNIT_ASSERT(!at_quad_edge.is_invalid());
+    CPPUNIT_ASSERT(at_quad_edge.at_corner());
+    CPPUNIT_ASSERT(!at_quad_edge.at_vertex());
+    CPPUNIT_ASSERT(at_quad_edge.at_edge());
+    CPPUNIT_ASSERT(!at_quad_edge.at_edge(0, 1));
+    CPPUNIT_ASSERT(at_quad_edge.at_edge(1, 2));
+    CPPUNIT_ASSERT_EQUAL(at_quad_edge.edge_vertices().first, (unsigned short)1);
+    CPPUNIT_ASSERT_EQUAL(at_quad_edge.edge_vertices().second, (unsigned short)2);
+    if (quad)
+    {
+      CPPUNIT_ASSERT(at_quad_edge.is_valid(*quad, 0.5 * (quad->point(1) + quad->point(2))));
+      for (const auto e : quad->edge_index_range())
+        CPPUNIT_ASSERT_EQUAL(at_quad_edge.at_edge(*quad, e), e == 1);
+
+      auto quad_edge_elem = at_quad_edge.build_edge(*quad);
+      CPPUNIT_ASSERT((quad_edge_elem->point(0).absolute_fuzzy_equals(quad->point(1))
+                      && quad_edge_elem->point(1).absolute_fuzzy_equals(quad->point(2))) ||
+                  ((quad_edge_elem->point(0).absolute_fuzzy_equals(quad->point(2))
+                       && quad_edge_elem->point(1).absolute_fuzzy_equals(quad->point(1)))));
+    }
+    const std::string at_edge_string = "at edge with vertices 1 and 2";
+    std::stringstream at_edge_ss;
+    at_edge_ss << at_quad_edge;
+    CPPUNIT_ASSERT_EQUAL(at_edge_string, at_quad_edge.print());
+    CPPUNIT_ASSERT_EQUAL(at_edge_string, at_edge_ss.str());
+#endif
+
+#if LIBMESH_DIM > 2
+    // HEX elem for edge testing
+    Mesh mesh_3d (*TestCommWorld);
+    MeshTools::Generation::build_cube (mesh_3d, 1, 1, 1);
+    const auto hex = mesh_3d.elem_ptr(0);
+
+    // At edge
+    const ElemCorner at_hex_edge(0, 1);
+    CPPUNIT_ASSERT(!at_hex_edge.is_invalid());
+    CPPUNIT_ASSERT(at_hex_edge.at_corner());
+    CPPUNIT_ASSERT(!at_hex_edge.at_vertex());
+    CPPUNIT_ASSERT(!at_hex_edge.at_vertex(0));
+    CPPUNIT_ASSERT(at_hex_edge.at_edge());
+    CPPUNIT_ASSERT(at_hex_edge.at_edge(0, 1));
+    CPPUNIT_ASSERT(at_hex_edge.at_edge(1, 0));
+    CPPUNIT_ASSERT_EQUAL(at_hex_edge.edge_vertices().first, (unsigned short)0);
+    CPPUNIT_ASSERT_EQUAL(at_hex_edge.edge_vertices().second, (unsigned short)1);
+    CPPUNIT_ASSERT(!at_hex_edge.is_invalid());
+    if (hex)
+    {
+      CPPUNIT_ASSERT(at_hex_edge.is_valid(*hex, (hex->point(0) + hex->point(1)) * 0.5));
+      for (const auto e : hex->edge_index_range())
+        CPPUNIT_ASSERT_EQUAL(at_hex_edge.at_edge(*hex, e), e == 0);
+      for (const auto v : hex->vertex_index_range())
+        CPPUNIT_ASSERT_EQUAL(at_hex_edge.is_valid(*hex, hex->point(v)), v == 0 || v == 1);
+      CPPUNIT_ASSERT(!at_hex_edge.is_valid(*hex, (hex->point(2)) * 0.5));
+
+      // build_edge
+      auto hex_edge_elem = at_hex_edge.build_edge(*hex);
+      CPPUNIT_ASSERT((hex_edge_elem->point(0).absolute_fuzzy_equals(hex->point(0))
+                      && hex_edge_elem->point(1).absolute_fuzzy_equals(hex->point(1))) ||
+                  ((hex_edge_elem->point(1).absolute_fuzzy_equals(hex->point(0))
+                       && hex_edge_elem->point(0).absolute_fuzzy_equals(hex->point(1)))));
+    }
+#endif // LIBMESH_DIM > 2
+
+    // For testing non-const methods
+    ElemCorner corner(0, 0);
+
+    // invalidate
+    corner.invalidate();
+    CPPUNIT_ASSERT(corner.is_invalid());
+
+    // set_vertex
+    corner.set_vertex(0);
+    CPPUNIT_ASSERT(corner.at_vertex(0));
+
+    // set_edge
+#if LIBMESH_DIM > 2
+    corner.set_edge(0, 1);
+    CPPUNIT_ASSERT(corner.at_edge(0, 1));
+    corner.set_edge(std::pair<unsigned short, unsigned short>(1, 2));
+    CPPUNIT_ASSERT(corner.at_edge(1, 2));
+#endif // LIBMESH_DIM > 2
+  }
+
+};
+#endif // LIBMESH_DIM > 1
+
+CPPUNIT_TEST_SUITE_REGISTRATION( ElemCornerTest );
diff --git a/tests/geom/elem_test.C b/tests/geom/elem_test.C
index 2795448558c..a4a20200004 100644
--- a/tests/geom/elem_test.C
+++ b/tests/geom/elem_test.C
@@ -488,6 +488,79 @@ public:
         CPPUNIT_ASSERT_EQUAL(elem->build_side_ptr(s)->type(), elem->side_type(s));
   }
 
+  void test_n_nodes_on_side()
+  {
+    LOG_UNIT_TEST;
+
+    for (const auto & elem : _mesh->active_local_element_ptr_range())
+      for (const auto s : elem->side_index_range())
+      {
+        unsigned int n_nodes_on_side = 0;
+        for (const auto n : elem->node_index_range())
+          if (elem->is_node_on_side(n, s))
+            ++n_nodes_on_side;
+        CPPUNIT_ASSERT_EQUAL(n_nodes_on_side, elem->n_nodes_on_side(s));
+      }
+  };
+
+  void test_n_nodes_on_edge()
+  {
+    LOG_UNIT_TEST;
+
+    for (const auto & elem : _mesh->active_local_element_ptr_range())
+      for (const auto e : elem->edge_index_range())
+      {
+        unsigned int n_nodes_on_edge = 0;
+        for (const auto n : elem->node_index_range())
+          if (elem->is_node_on_edge(n, e))
+            ++n_nodes_on_edge;
+        CPPUNIT_ASSERT_EQUAL(n_nodes_on_edge, elem->n_nodes_on_edge(e));
+      }
+  };
+
+  void test_n_vertices_on_side()
+  {
+    LOG_UNIT_TEST;
+
+    for (const auto & elem : _mesh->active_local_element_ptr_range())
+      for (const auto s : elem->side_index_range())
+      {
+        unsigned int n_vertices_on_side = 0;
+        for (const auto v : elem->vertex_index_range())
+          if (elem->is_node_on_side(v, s))
+            ++n_vertices_on_side;
+        CPPUNIT_ASSERT_EQUAL(n_vertices_on_side, elem->n_vertices_on_side(s));
+      }
+  };
+
+  void test_nodes_on_side_ptr()
+  {
+    LOG_UNIT_TEST;
+
+    for (const auto & elem : _mesh->active_local_element_ptr_range())
+      for (const auto s : elem->side_index_range())
+      {
+        const auto nodes_on_side = elem->nodes_on_side(s);
+        const auto nodes_on_side_ptr = elem->nodes_on_side_ptr(s);
+        for (const auto i : index_range(nodes_on_side))
+          CPPUNIT_ASSERT_EQUAL(nodes_on_side[i], nodes_on_side_ptr[i]);
+      }
+  };
+
+  void test_nodes_on_edge_ptr()
+  {
+    LOG_UNIT_TEST;
+
+    for (const auto & elem : _mesh->active_local_element_ptr_range())
+      for (const auto s : elem->side_index_range())
+      {
+        const auto nodes_on_edge = elem->nodes_on_edge(s);
+        const auto nodes_on_edge_ptr = elem->nodes_on_edge_ptr(s);
+        for (const auto i : index_range(nodes_on_edge))
+          CPPUNIT_ASSERT_EQUAL(nodes_on_edge_ptr[i], nodes_on_edge_ptr[i]);
+      }
+  };
+
   void test_elem_side_builder()
   {
     LOG_UNIT_TEST;
@@ -521,6 +594,11 @@ public:
   CPPUNIT_TEST( test_contains_point_node );     \
   CPPUNIT_TEST( test_center_node_on_side );     \
   CPPUNIT_TEST( test_side_type );               \
+  CPPUNIT_TEST( test_n_nodes_on_side );         \
+  CPPUNIT_TEST( test_n_nodes_on_edge );         \
+  CPPUNIT_TEST( test_n_vertices_on_side );      \
+  CPPUNIT_TEST( test_nodes_on_side_ptr );       \
+  CPPUNIT_TEST( test_nodes_on_edge_ptr );       \
   CPPUNIT_TEST( test_elem_side_builder );
 
 #define INSTANTIATE_ELEMTEST(elemtype)                          \
diff --git a/tests/geom/intersection_tools_test.C b/tests/geom/intersection_tools_test.C
new file mode 100644
index 00000000000..b71995b4248
--- /dev/null
+++ b/tests/geom/intersection_tools_test.C
@@ -0,0 +1,328 @@
+#include "libmesh_cppunit.h"
+
+#include "test_comm.h"
+
+#include <libmesh/intersection_tools.h>
+#include <libmesh/point.h>
+#include <libmesh/int_range.h>
+#include <libmesh/enum_elem_type.h>
+#include <libmesh/mesh.h>
+#include <libmesh/mesh_generation.h>
+#include <libmesh/elem.h>
+#include <libmesh/elem_corner.h>
+
+using namespace libMesh;
+
+class IntersectionToolsTest : public CppUnit::TestCase
+{
+
+public:
+  LIBMESH_CPPUNIT_TEST_SUITE( IntersectionToolsTest );
+  CPPUNIT_TEST( within_segment );
+  CPPUNIT_TEST( collinear );
+  CPPUNIT_TEST_SUITE_END();
+
+public:
+
+  void within_segment()
+  {
+    LOG_UNIT_TEST;
+
+    const std::vector<Real> scales{1.5e-7, 1, 1.5e7};
+    for (const auto scale : scales)
+    {
+      const auto s1 = scale * Point(1.0, 2.0, 3.0);
+      const auto s2 = scale * Point(2.0, 3.0, 4.0);
+
+      const auto length_vec = s2 - s1;
+      const auto length = length_vec.norm();
+      const auto s1_to_s2 = length_vec / length;
+
+      int segments = 10;
+      Real dx = (Real)1 / segments * length;
+      for (const auto i : make_range(-1, segments + 1))
+      {
+        const auto p = s1 + (Real)i * dx * s1_to_s2;
+        IntersectionTools::WithinSegmentResult within_result = IntersectionTools::WithinSegmentResult::NOT_WITHIN;
+        if (i == 0)
+          within_result = IntersectionTools::WithinSegmentResult::AT_BEGINNING;
+        else if (i > 0 && i < segments)
+          within_result = IntersectionTools::WithinSegmentResult::BETWEEN;
+        else if (i == segments)
+          within_result = IntersectionTools::WithinSegmentResult::AT_END;
+
+        CPPUNIT_ASSERT_EQUAL(IntersectionTools::within_segment(s1, s2, p), within_result);
+      }
+
+      CPPUNIT_ASSERT_EQUAL(IntersectionTools::within_segment(s1, s2, scale * Point(9.9, 5, 3)),
+                          IntersectionTools::WithinSegmentResult::NOT_WITHIN);
+    }
+  }
+
+  void collinear()
+  {
+    LOG_UNIT_TEST;
+
+    const auto do_assert = [](const Point & p1,
+                              const Point & p2,
+                              const Point & p3,
+                              const bool collinear)
+    {
+      const std::vector<Real> scales{1e-8, 1, 1e8};
+      for (const auto scale : scales)
+      {
+        const auto p1s = scale * p1;
+        const auto p2s = scale * p2;
+        const auto p3s = scale * p3;
+        CPPUNIT_ASSERT_EQUAL(IntersectionTools::collinear(p1s, p2s, p3s), collinear);
+        CPPUNIT_ASSERT_EQUAL(IntersectionTools::collinear(p1s, p3s, p2s), collinear);
+        CPPUNIT_ASSERT_EQUAL(IntersectionTools::collinear(p2s, p1s, p3s), collinear);
+        CPPUNIT_ASSERT_EQUAL(IntersectionTools::collinear(p2s, p3s, p1s), collinear);
+        CPPUNIT_ASSERT_EQUAL(IntersectionTools::collinear(p3s, p1s, p2s), collinear);
+        CPPUNIT_ASSERT_EQUAL(IntersectionTools::collinear(p3s, p2s, p1s), collinear);
+      }
+    };
+
+    // two of the same points
+    do_assert(Point(1, 2, 3), Point(1, 2, 3), Point(4, 5, 6), true);
+
+    // three of the same points
+    do_assert(Point(1, 2, 3), Point(1, 2, 3), Point(1, 2, 3), true);
+
+    // all in a line
+    do_assert(Point(1, 1, 2), Point(1, 1, 3), Point(1, 1, 0), true);
+
+    // not collinear
+    do_assert(Point(0, 1, 2), Point(0, 1, 3), Point(1, 5, 10), false);
+  }
+};
+
+CPPUNIT_TEST_SUITE_REGISTRATION( IntersectionToolsTest );
+
+template <ElemType elem_type>
+class MeshedIntersectionToolsTest : public CppUnit::TestCase {
+
+private:
+  std::vector<std::unique_ptr<Mesh>> _meshes;
+
+protected:
+  std::string libmesh_suite_name;
+
+public:
+  void setUp()
+  {
+    const std::vector<Real> scales{1e-8, 1, 1e8};
+    for (const auto scale : scales)
+    {
+      const Real minpos = scale * 1.5;
+      const Real maxpos = scale * 4.86;
+      const unsigned int N = 3;
+
+      std::unique_ptr<Mesh> mesh = std::make_unique<Mesh>(*TestCommWorld);
+      auto test_elem = Elem::build(elem_type);
+
+      const unsigned int dim = test_elem->dim();
+      const unsigned int use_y = dim > 1;
+      const unsigned int use_z = dim > 2;
+
+      MeshTools::Generation::build_cube (*mesh,
+                                        N, N*use_y, N*use_z,
+                                        minpos, maxpos,
+                                        minpos, use_y*maxpos,
+                                        minpos, use_z*maxpos,
+                                        elem_type);
+
+      _meshes.emplace_back(std::move(mesh));
+    }
+  }
+
+  void test_within_edge()
+  {
+    LOG_UNIT_TEST;
+
+    // check locations at every node
+    for (const auto & mesh : _meshes)
+      for (const auto & elem : mesh->active_local_element_ptr_range())
+        for (const auto e : elem->edge_index_range())
+          for (const auto n : elem->nodes_on_edge(e))
+          {
+            ElemCorner corner;
+            const auto within = IntersectionTools::within_edge(*elem, elem->point(n), corner);
+            CPPUNIT_ASSERT(within);
+
+            if (mesh->mesh_dimension() == 3)
+              for (const auto s : elem->side_index_range())
+              {
+                ElemCorner corner_on_side;
+                const auto within_on_side = IntersectionTools::within_edge_on_side(*elem,
+                                                                                  elem->point(n),
+                                                                                  s,
+                                                                                  corner_on_side);
+
+                const auto node_is_on_side = elem->is_node_on_side(n, s);
+                CPPUNIT_ASSERT_EQUAL(within_on_side, node_is_on_side);
+                if (node_is_on_side)
+                {
+                  CPPUNIT_ASSERT_EQUAL(corner.at_vertex(n), corner_on_side.at_vertex(n));
+                  CPPUNIT_ASSERT_EQUAL(corner.at_edge(*elem, e), corner_on_side.at_edge(*elem, e));
+                  CPPUNIT_ASSERT_EQUAL(elem->is_vertex(n), corner_on_side.at_vertex(n));
+                  CPPUNIT_ASSERT_EQUAL(elem->is_vertex(n), !corner_on_side.at_edge(*elem, e));
+                }
+              }
+          }
+
+    // cut edges into segments
+    for (const auto & mesh : _meshes)
+      for (const auto & elem : mesh->active_local_element_ptr_range())
+        for (const auto e : elem->edge_index_range())
+        {
+          const auto nodes_on_edge = elem->nodes_on_edge(e);
+          const auto & p1 = elem->point(nodes_on_edge[0]);
+          const auto & p2 = elem->point(nodes_on_edge[1]);
+          const auto length_vec = p2 - p1;
+          const auto length = length_vec.norm();
+          const auto p1_to_p2 = length_vec / length;
+
+          int segments = 5;
+          Real dx = (Real)1 / segments * length;
+          for (const auto i : make_range(-1, segments + 1))
+          {
+            const auto p = p1 + Real(i) * dx * p1_to_p2;
+
+            ElemCorner corner;
+            const auto within = IntersectionTools::within_edge(*elem, p, corner);
+            CPPUNIT_ASSERT_EQUAL(within, i >= 0 && i <= segments);
+
+            if (mesh->mesh_dimension() == 3)
+              for (const auto s : elem->side_index_range())
+                if (elem->is_edge_on_side(e, s))
+                {
+                  ElemCorner corner_on_side;
+                  const auto within_on_side = IntersectionTools::within_edge_on_side(*elem,
+                                                                                    p,
+                                                                                    s,
+                                                                                    corner_on_side);
+
+                  CPPUNIT_ASSERT_EQUAL(within_on_side, within);
+                  CPPUNIT_ASSERT_EQUAL(corner_on_side.at_vertex(nodes_on_edge[0]), i == 0);
+                  CPPUNIT_ASSERT_EQUAL(corner_on_side.at_vertex(nodes_on_edge[1]), i == segments);
+                  CPPUNIT_ASSERT_EQUAL(corner_on_side.at_edge(*elem, e), i > 0 && i < segments);
+                }
+          }
+        }
+
+    // check elem centroids
+    for (const auto & mesh : _meshes)
+      for (const auto & elem : mesh->active_local_element_ptr_range())
+      {
+        const auto p = elem->vertex_average();
+
+        ElemCorner corner;
+        CPPUNIT_ASSERT(!IntersectionTools::within_edge(*elem, p, corner));
+
+        if (mesh->mesh_dimension() == 3)
+          for (const auto s : elem->side_index_range())
+          {
+            ElemCorner corner_on_side;
+            CPPUNIT_ASSERT(!IntersectionTools::within_edge_on_side(*elem, p, s, corner_on_side));
+          }
+      }
+  }
+
+  void test_at_vertex()
+  {
+    LOG_UNIT_TEST;
+
+    // check locations at every node
+    for (const auto & mesh : _meshes)
+      for (const auto & elem : mesh->active_local_element_ptr_range())
+        for (const auto n : elem->node_index_range())
+        {
+          const auto & p = elem->point(n);
+          const auto is_vertex = elem->is_vertex(n);
+          const auto at_vertex = IntersectionTools::at_vertex(*elem, p);
+
+          if (is_vertex)
+            CPPUNIT_ASSERT_EQUAL(at_vertex, (unsigned short)n);
+          else
+            CPPUNIT_ASSERT_EQUAL(at_vertex, Elem::invalid_vertex);
+
+          for (const auto s : elem->side_index_range())
+          {
+            const auto at_vertex_on_side = IntersectionTools::at_vertex_on_side(*elem, p, s);
+            if (is_vertex && elem->is_node_on_side(n, s))
+              CPPUNIT_ASSERT_EQUAL(at_vertex_on_side, at_vertex);
+            else
+              CPPUNIT_ASSERT_EQUAL(at_vertex_on_side, Elem::invalid_vertex);
+          }
+        }
+
+    // check edge centroids
+    for (const auto & mesh : _meshes)
+      if (mesh->mesh_dimension() > 1)
+        for (const auto & elem : mesh->active_local_element_ptr_range())
+          for (const auto e : elem->edge_index_range())
+          {
+            const auto edge = elem->build_edge_ptr(e);
+            const auto at_vertex = IntersectionTools::at_vertex(*elem, edge->vertex_average());
+            CPPUNIT_ASSERT_EQUAL(at_vertex, Elem::invalid_vertex);
+          }
+
+    // check elem centroids
+    for (const auto & mesh : _meshes)
+      for (const auto & elem : mesh->active_local_element_ptr_range())
+        CPPUNIT_ASSERT_EQUAL(IntersectionTools::at_vertex(*elem, elem->centroid()), Elem::invalid_vertex);
+  }
+
+};
+
+#define MESHEDINTERSECTIONTOOLSTEST  \
+  CPPUNIT_TEST( test_within_edge );  \
+  CPPUNIT_TEST( test_at_vertex );
+
+#define INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(elemtype)                                       \
+  class MeshedIntersectionToolsTest_##elemtype : public MeshedIntersectionToolsTest<elemtype> { \
+  public:                                                                                       \
+  MeshedIntersectionToolsTest_##elemtype() :                                                    \
+    MeshedIntersectionToolsTest<elemtype>() {                                                   \
+    if (unitlog->summarized_logs_enabled())                                                     \
+      this->libmesh_suite_name = "MeshedIntersectionToolsTest";                                 \
+    else                                                                                        \
+      this->libmesh_suite_name = "MeshedIntersectionToolsTest_" #elemtype;                      \
+  }                                                                                             \
+  CPPUNIT_TEST_SUITE( MeshedIntersectionToolsTest_##elemtype );                                 \
+  MESHEDINTERSECTIONTOOLSTEST;                                                                  \
+  CPPUNIT_TEST_SUITE_END();                                                                     \
+  };                                                                                            \
+                                                                                                \
+  CPPUNIT_TEST_SUITE_REGISTRATION( MeshedIntersectionToolsTest_##elemtype )
+
+#if LIBMESH_DIM > 1
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(TRI3);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(TRI6);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(TRI7);
+
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(QUAD4);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(QUAD8);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(QUAD9);
+#endif // LIBMESH_DIM > 1
+
+#if LIBMESH_DIM > 2
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(TET4);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(TET10);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(TET14);
+
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(HEX8);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(HEX20);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(HEX27);
+
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(PRISM6);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(PRISM15);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(PRISM18);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(PRISM20);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(PRISM21);
+
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(PYRAMID5);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(PYRAMID13);
+INSTANTIATE_MESHEDINTERSECTIONTOOLSTEST(PYRAMID14);
+#endif // LIBMESH_DIM > 2