+ A rotation of a vector is a change which only
+ alters the direction, not the length, of a vector. A
+ rotation consists of a rotation axis and a
+ rotation rate. By taking the rotation axis as a
+ direction and the rotation rate as a length, we can write
+ the rotation as a vector, known as the angular velocity
+ vector
+ + +
+
+ Rotation axis:
+
+ Angular velocity vector \(\vec\omega\). The direction of + \(\vec\omega\) is the axis of rotation, while the magnitude + is the speed of rotation (positive direction given by the + right-hand rule). +
++ The Greek letter ω (lowercase omega) is + the last letter of the Greek + alphabet, leading to expressions such as “from alpha + to omega” meaning “from start to end”. Omega literally + means O-mega, meaning O-large, as capital Omega (written + Ω) developed from capital Omicron + (written Ο) by breaking the circle and turning up the + edges. Omicron is literally O-micron, meaning O-small, and + it is the ancestor of the Latin + letter O that we use today in English. +
+| Α | α | alpha | +Ι | ι | iota | +Ρ | ρ | rho | +
| Β | β | beta | +Κ | κ | kappa | +Σ | σ | sigma | +
| Γ | γ | gamma | +Λ | λ | lambda | +Τ | τ | tau | +
| Δ | δ | delta | +Μ | μ | mu | +Υ | υ | upsilon | +
| Ε | ε | epsilon | +Ν | ν | nu | +Φ | φ | phi | +
| Ζ | ζ | zeta | +Ξ | ξ | xi | +Χ | χ | chi | +
| Η | η | eta | +Ο | ο | omicron | +Ψ | ψ | psi | +
| Θ | θ | theta | +Π | π | pi | +Ω | ω | omega | +
+ The Greek alphabet, shown above, was the first true alphabet, + meaning that it has letters representing phonemes + (basic significant sounds) and includes vowels as + well as consonants. The + Greek alphabet was was derived from the earlier Phoenician + alphabet, which was probably the original parent of + all alphabets. This shows that the idea of an alphabet is + so non-obvious that it has only ever been invented once, + and then always copied after that. +
+
+ If a unit vector
+ Using the same approach as #rvc-em we write
+ This can be seen from the fact that
+ In 2D the angular velocity can be thought of as a scalar + (positive for counter-clockwise, negative for clockwise). + This scalar is just the out-of-plane component of the + full angular velocity vector. We can draw the angular + velocity as either a vector pointing out of the plane, or + as a circle-arrow in the plane, which is simpler for 2D + diagrams. +
+ ++ + +
++ Show: + +
++ Comparison of the vector and scalar representations of + \(\vec\omega\) for 2D rotations. +
++ In 2D the angular velocity scalar \(\omega\) is simply the + derivative of the rotation angle \(\theta\) in the plane: +
+ +
+ Take
+ The right-hand rule convention for angular velocities means + that counter-clockwise rotations are positive, just like the + usual angle direction convention. +
+ ++ Angular directions have long been considered to have + magical or spiritual significance. In Britain the + counterclockwise direction was once known + as widdershins, + and it was considered unlucky to travel around a church in + a widdershins direction. +
++ Interestingly, right-handed people tend to naturally draw + circles in a counterclockwise direction, and clockwise + drawing in right-handed children is an early warning sign + for the later development of schizophrenia [Blau, 1977]. +
+References
+-
+
- + T. H. Blau. + Torque and schizophrenic vulnerability. + American Psychologist, 32(12):997–1005, 1977. + DOI: 10.1037/0003-066X.32.12.997. + +
+ The fact that vectors don't have + positions means that vector rotations are independent of + where vectors are drawn, just like for derivatives. +
+ ++ + +
++ Show: + + + +
++ Rotational motion of vectors which are drawn moving + about. Note that the drawn position does not affect the + angular velocity $\omega$ or the derivative vectors. +
+
+ Rotations are rigid transformations, meaning that
+ they keep constant all vector lengths and all relative
+ vector angles. These facts are reflected in the following
+ results, which all consider two vectors
+ Using #rkr-el and the scalar
+ triple product formula #rvi-es gives:
+
+
+ We first consider the dot product
+ From #rkr-el we know that the
+ derivative is
+
+
+ Rodrigues’ rotation formula gives an explicit formula for a + vector rotated by an angle about a given axis. +
+ +
+ Assume
+
+ Now we want to convert from the
+
+
+ Substituting these into the rotated vector expression + above gives +
+
+
+ Consider a particle moving with position vector
+ These equations are definitions of the basis vectors, so
+ the only thing to derive is the alternative formula for
+
+ The tangential basis vector
+ + + +
+| Show: | ++ + + + + | +
| + | + + + + + | +
+ Tangential/normal basis associated with movement around a
+ curve in 3D. Observe that the velocity
+ As the point \(P\) moves along its path, the associated + tangential/normal basis rotates with an angular velocity + vector \(\omega\) given by: +
+ +
+ We start by writing the angular velocity in the
+ tangential/normal basis, giving:
+
+
+ Knowing the angular velocity vector of the tangential/normal + basis allows us to easily compute the time derivatives of + each tangential/normal basis vector, as follows: +
+ +
+ We can use the expression #rkt-ew
+ for
+ The tangential/normal basis is also called the
+ Frenet–Serret frame after Jean
+ Frédéric Frenet and Joseph
+ Alfred Serret, who discovered it independently around
+ 1850. The equations #rkt-ed for the
+ basis derivatives are often called the Frenet-Serret
+ formulas, typically written in terms of \(s\) derivatives:
+
+
+ If we divide the angular velocity vector #rkt-ew by \(v\) then we obtain the
+ vector
+
+
+ While the motion of a point \(P\) along a path defines the + tangential/normal basis, we can also use this basis to + express the kinematics of \(P\) itself, giving the following + expressions for velocity and acceleration. +
+ +
+ From the definition #rkt-eb of
+
+ The above formula shows that the normal acceleration + component \(a_n\) is determined by the radius of curvature. We + can therefore also find the radius of curvature from knowing + the normal acceleration: +
+ +
+ From #rkt-ev we known that the
+ normal component of the acceleration is given by
+ Now because
+ + + +
+| Movement: | +circle | +var-circle | +ellipse | +arc | +
| + | trefoil | +eight | +comet | +pendulum | +
| Show: | ++ + + + + | +|||
| + | + + + + + | +|||
| Origin: | +\(O_1\) | +\(O_2\) | ++ | + |
+ Velocity and acceleration in the tangential/normal + basis. Note that the tangential/normal basis does not + depend on the choice of origin or the position vector, in + contrast to the polar basis. +
++ To better understand the geometry of the tangential/normal + basis, we can use the curvature \(\kappa\) to + describe the curving of the path, and the torsion + \(\tau\) to describe the rotation of the basis about the + path. These quantities are defined by: +
+ ++ These equations are definitions, so we need only check + that the two expressions for each of \(\kappa\) and \(\tau\) + are equivalent. This follows immediately, however, from + the chain-rule conversions #rkt-ea + between \(d/ds\) and \(d/dt\). +
++ The radius of curvature \(\rho\) is the radius of + equivalent circular motion, and the torsion + determines the rate of rotation of the osculating plane, as + described below in Section #rkt-so. +
+ ++ Given a parametric curve, its curvature can be directly + evaluated with: +
+ +
+ Because the tangential/normal basis vectors #rkt-eb are all normalized, it does
+ not matter how fast the point moves along the line. We
+ can thus evaluate the parametric curve as a function of
+ time, giving
+ While the above formula can be used in 2D by taking the + third component to be zero, it can also be written in an + explicitly 2D form: +
+ +
+ This equation is just #rkt-ec
+ written in explicit 2D coordinates. To see this, we
+ first take the position vector
+ We can take this a step further, and obtain an expression for an explicitly defined function. +
+ +
+ Use equation #rkt‑e2 and parametrize the curve using \(x\) as the parametrization variable. Namely:
+
+
@@ -372,8 +372,8 @@ import Col from "../../components/Col.astro"
Time:
-
As you can see, while a superscribed dot always denotes @@ -490,7 +490,7 @@ import Col from "../../components/Col.astro"
@@ -512,6 +512,80 @@ import Col from "../../components/Col.astro"
+ In a fixed basis we differentiate a vector by + differentiating each component: +
+ +
+ Writing a time-dependent vector expression in a fixed basis gives:
+
+
+ When we differentiate a vector by differentiating each + component and leaving the basis vectors unchanged, we + are assuming that the basis vectors themselves are not + changing with time. If they are, then we need to take this into account as + well. +
++ + +
+| Time: | ++ + \(t = \) 0 s + | +
| Show: | ++ + + + | +
| Basis: | +
+
+ |
+
+ The vector derivative decomposed into components. This + demonstrates graphically that each component of a vector + in a particular basis is simply a scalar function, and the + corresponding derivative component is the regular scalar + derivative. +
+