Express rotation of the plane about a point as the product of a translation and a rotation about the origin

Question

I need help with the following problem:

a) Let $s$ be a rotation of the plane with angle $\frac{\pi}{2}$ about the point $(1,1)^t$. Write the formula for $s$ as a product $t_a \rho _\theta $

Edit: $t_a$ is a translation and $p_\theta$ is a rotation about the origin.

Answer 1

a) $t_a$ does not rotate lines, but $s$ rotates them by $\frac{\pi}{2}$, so $\theta$ must be $\frac{\pi}{2}$. Also, $s$ sends $(0,0)$ to $(2,0)$, while $\rho_\theta$ fixes the origin, so $a$ must be $(2,0)$.

b) what is $r$?

Answer 2

This is a straightforward calculation if you use homogeneous coordinates, which allow affine transformations of the plane to be represented as $3\times3$ matrices. Since the center of rotation is given as a column vector, I’ll assume that application of a transformation is left-multiplication by a matrix. The computations are similar if you’re right-multiplying row vectors by the matrices instead.

The desired transformation can be achieved by translating to the origin, rotating by $\frac\pi2$, then translating back: $$ \pmatrix{1&0&1\\0&1&1\\0&0&1}\pmatrix{0&-1&0\\1&0&0\\0&0&1}\pmatrix{1&0&-1\\0&1&-1\\0&0&1}=\pmatrix{0&-1&2\\1&0&0\\0&0&1}. $$We’re asked to express this as the product of a translation and rotation. The transformation maps the positive $x$-axis onto a ray parallel to the positive $y$-axis, so the rotation must be through an angle of $\frac\pi2$. Right-multiplying the above matrix by a rotation through $-\frac\pi2$ gives $$ \pmatrix{0&-1&2\\1&0&0\\0&0&1}\pmatrix{0&1&0\\-1&0&0\\0&0&1}=\pmatrix{1&0&2\\0&1&0\\0&0&1}, $$ which is indeed a translation, as required. Thus, the transformation is equivalent to $t_{(2,0)}\rho_{\pi/2}$.