Point reflection ($x\mapsto 2c-x$)

A point reflection about $c$ sends each point $x$ to $2c-x$. The point $c$ is the midpoint of $x$ and its image, and applying the map twice is the identity. It is a Euclidean isometry since $(2c-x)-(2c-y)=-(x-y)$ preserves the Euclidean norm. In the Euclidean plane it is a rotation through $\pi$; in dimension $n$ its determinant is $(-1{)}^n$, so it reverses orientation exactly in odd dimension. A figure invariant under this map has $c$ as a center of symmetry.

A point $\mathbf{c}$ is a center of symmetry of a set $S$ in Euclidean space if point reflection $\mathbf{x}\mapsto 2\mathbf{c}-\mathbf{x}$ maps $S$ onto itself. Every ellipse has its center of an ellipse as a center of symmetry. More generally, a nonempty bounded set has at most one: two such reflections compose to translation by twice the difference of their centers, and iterating a nonzero translation contradicts boundedness.