Consider a circular wheel with radius $r$ moving to the right along the ground at a constant velocity. The ground is the x-axis, and the wheel moves in the positive direction. Assume that $Q_0$ is at the origin, so that the position vector of $Q_0$ is the zero vector. Let $Q_t$ be the position of the point after t seconds ($Q_t$ always stays on the boundary of the circle for all $t$ and the point moves in clockwise-direction), and let $P_t$ be the position of the centre of the wheel after t seconds, for all $t ≥ 0.$ I want to find
i) $\sigma \in \text {Bij}(\mathbb R^2) $ such that $P_{t+1} = \sigma (P_{t}) \,\,\, \forall t \geq 0.$
ii) $\tau \in \text {Bij}(\mathbb R^2) $ such that $Q_{t+1} = \tau (Q_{t}) \,\,\, \forall t \geq 0.$
Here Bij($\mathbb R^2$) is the set of all transformations on $\mathbb R^2.$
My answers:
i) $\sigma$ is the map $T_{A,\mathbb b} : \mathbb R^2 \to \mathbb R^2$, $\mathbb x \mapsto A\mathbb x \,\, + \dfrac{2\pi r}{T}t $ where $A$ is the clockwise rotation matrix and $\mathbb b = \dfrac{2\pi r}{T}t $.
ii) I want to express $\tau$ as a product of transformations, defined geometrically. The definition of $\tau$ will certainly involve $P_t.$ But I don't know how to do it.
Any help appreciated.
The wheel is moving at a constant velocity, let's call it $v$. In one unit of time it changes $x$ by $v\cdot 1$, while keeping $y$ fixed, $y=r$. So $$(x_{t+1},y_{t+1})=(x_t+v,y_t)=(x_t,y_t)+(v,0)$$ The point of the wheel rotates around the axis at a constant angular speed. To describe the point in the fixed reference frame, you need to account for the fact that the axis is moving as well, so you would need to add the motion of the center. In the reference frame at the center of the axis $$x_t=-r\sin\omega t\\y_t=r(1-\cos \omega t)$$ When you add the motion of the center, $$x_t=vt-r\sin\omega t\\y_t=r(1-\cos \omega t)$$. Here $\omega$ is the angular speed. If the wheel rotates without slipping, $$v=\omega r$$
Can you finish from here to write $\tau$? Write $x_{t+1}$ and $y_{t+1}$ in terms of the quantities at $t$.
Note that you are describing the cycloid motion, in case you want to look it up.