Reflection (mathematics)

A reflection fixes a hyperplane pointwise and reverses its normal direction.

A plane reflection with unit normal $n$ is $I-2n{n}^T$; it fixes the plane $n^{\perp }$ and negates $n$.

A planar rotation about a point on the axis of a Euclidean reflection, composed after that Euclidean reflection, is a Euclidean reflection whose axis has been rotated by half the rotation angle. In coordinates centred at the fixed point, with the original axis horizontal, write $S=\mathrm{diag}(1,-1)$. ThenThis matrix fixes the direction $(\cos (\alpha /2),\sin (\alpha /2))$ and reverses its perpendicular direction. The axis is an unoriented line, so its angle is defined modulo $\pi$.

For a nonzero normal vector $a$, reflection in $a\cdot x=0$ has matrixIts eigenvalues are $-1$ along $a$ and $1$ on $a^{\perp }$, so its determinant is $-1$.

The composition of reflections in two planes through the origin is a rotation about their line of intersection through twice the oriented angle between the planes.