# Reflection Symmetry

I’m interested in a method for determining if a simple polygon has reflection symmetry.

Let $r$ denote reflection about the y-axis and $\rho_\theta$ denote counter clockwise rotation by the angle $\theta$. Note that P has reflection symmetry iff $r \rho_\theta P = P$ for some angle $\theta$. We also know $r \rho_\theta = \rho_{-\theta} r$. Hence $r\rho_{\theta} P = \rho_{-\theta} r P = P$. Multiply both sides by $\rho_\theta$ to have $rP = \rho_{\theta} P$.

(1) Thus P has reflection symmetry iff $rP = \rho_{\theta}P$ for some angle $\theta$.

This simple fact enables a easy method to determine if P has reflection symmetry. Take a simple polygon P. First assume the centroid of P is the origin this is needed to ensure that a rotation about the x-axis is a rotation of P and not a rotation and translation of P.

From (1) we know that $rP = \rho_\theta P$ for some angle $\theta$. To determine $\theta$ we pick a vertex, v, of $P$. We know that v is rotated into some vertex w of $rP$. Hence we try rotating v into every vertex w of $rP$ until we get a match.