# Composition of two pivot-angle rotations in 2D

Suppose I have a set of points $P$ in the 2D plane.

Let $(\theta_1, p_1)$ and $(\theta_2, p_2)$ be two axis-pivot rotation, where for each $i \in \{1, 2\}$, $\theta_i$ is a rotation angle and $p_i$ the pivot point of this rotation. I would like to rotate each point of $P$ by an angle of $\theta_1$ around $p_1$, then rotate each resulting point by an angle of $\theta_2$ around $p_2$.

The resulting location of each point of $P$ can easily be computed by calculating the result of the first rotation, then the second.

However, can this be described by a single axis-pivot rotation $(\theta_3, P_3)$?

Semi-related is this question: Composition of two axis-angle rotations

However, it applies to 3D and seems more complicated than it needs to be.

• I don't know the answer, but if it can be, then the pivot is a fixed point of the composition, so I would first try to find such a fixed point. Feb 27 '18 at 19:45
• If it was possible to describe it by a single rotation, then the involute of , e.g., a triangle would be a circle. It might be possible for infinitesimal rotations (shall think about). Feb 27 '18 at 19:54