Composition of Rotations

Excerpts

• Composition of Rotations

  A version refers to the change of direction of a line, but a rotation refers to a rigid body. The composition of rotations is a different matter from the composition of versions.

• Composition of Rotations

  Suppose that a rigid body rotates $\theta$ radians round the axis $\beta$ passing through the point $O$, and that $R$ is the radius vector from $O$ to some particle.

• Composition of Rotations

  The expression $(\beta^\frac{b}{2}\gamma^\frac{c}{2})^2$ is not, as might be supposed, identical with $\beta^b\gamma^c$. The former reduces to the latter only when $\beta$ and $\gamma$ are the same or opposite.

• Composition of Rotations

  When $b$ and $c$ are infinitesimals, $\cos\beta^b \times \gamma^c = 1$, and $\Sin \beta^b \times \gamma^c = b \cdot \beta + c \cdot \gamma$, which is the parallelogram rule for the composition of infinitesimal rotations.

• Composition of Rotations

  It indicates that quaternion multiplication in the most general sense has its physical meaning in the composition of rotations.

Equations

• Composition of Rotations

  \beta^\theta R = \mathrm{S}\beta R \cdot \beta + \cos \theta(\mathrm{V}\beta R)\beta + \sin \theta \mathrm{V}\beta R

  A line R is turned through angle theta about the axis beta; the result is the scalar part of beta R times beta, plus cos theta times the vector (V beta R) beta, plus sin theta times V beta R.

• Composition of Rotations

  \beta^\theta R = \cos \theta R + \sin \theta \mathrm{V}(\beta R)

  When the radius vector is perpendicular to the axis, a rotation through theta gives cos theta times R plus sin theta times the vector part of beta R.

• Composition of Rotations

  \mathrm{S}\beta R = lx + my + nz

  The scalar part of beta R equals the sum of the products of the direction cosines of beta with the coordinates of R.

• Composition of Rotations

  l^2 + m^2 + n^2 = 1

  The components of the axis beta are direction cosines, so their squares sum to one.

• Composition of Rotations

  \beta^b\rho = \beta^\frac{-b}{2}\rho^\frac{\pi}{2}\beta^\frac{b}{2}

  The versor rotating a line by angle b about beta equals the product of a half-angle versor, a quadrantal versor and the opposite half-angle versor.

• Composition of Rotations

  e^{-\frac{1}{2}b\beta^\frac{\pi}{2} + \frac{1}{2}\pi\rho^\frac{\pi}{2} + \frac{1}{2}b\beta^\frac{\pi}{2}}

  The versor beta^(-b/2) rho^(pi/2) beta^(b/2) is written as an exponential of a trinomial.

• Composition of Rotations

  \beta^b \times \gamma^c = (\beta^\frac{b}{2}\gamma^\frac{c}{2})^2

  The single rotation equivalent to rotation b about beta followed by rotation c about gamma is the square of the product of the two half-angle versors.

• Composition of Rotations

  \cos\frac{b}{2}\,\cos\frac{c}{2}-\sin\frac{b}{2}\,\sin\frac{c}{2}

  The cosine of the product of the two half-angle versors is cos(b/2) cos(c/2) minus sin(b/2) sin(c/2); this quantity is called m.

• Composition of Rotations

  \beta^b \times \gamma^c = m^2 - n^2 + 2mn \cdot \nu

  The composed rotation has cosine m^2 minus n^2, and its directed sine is 2mn times the unit vector nu.

• Composition of Rotations

  \cos\beta^b \times \gamma^c = 1

  For infinitesimal rotations the cosine of the composed rotation is 1.

• Composition of Rotations

  \Sin \beta^b \times \gamma^c = b \cdot \beta + c \cdot \gamma

  For infinitesimal rotations the directed sine of the composed rotation is the sum of the two infinitesimal rotation vectors, which is the parallelogram rule.

Problems