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.

A rotation $R(\theta,P)$ can be decomposed as the product of two reflections $P_r$ and $P_s$, about any two lines $r$ and $s$ intersecting at $P$ and forming an angle $\theta/2$ between them: $R(\theta,P)=P_s\circ P_r$.
If you have two rotations  $R(\theta_1,P_1)$, $R(\theta_2,P_2)$, and $r$ is line $P_1P_2$, you can then choose other two lines $a$, $b$ such that
$$
R(\theta_1,P_1)=P_r\circ P_{a},\quad R(\theta_2,P_2)=P_{b}\circ P_r.
$$
It follows that
$$
R(\theta_2,P_2)\circ R(\theta_1,P_1)=P_{b}\circ P_r\circ P_r\circ P_{a}=
P_{b}\circ P_{a}.
$$
If lines $a$ and $b$ intersect at $P_3$, this is indeed a rotation. If instead $a$ and $b$ are parallel, this is a translation: that happens if $\theta_1+\theta_2$ is a multiple of 360°.
