Reflection in l

Reflection in the line l is the transformation of the plane that maps each point P to its mirror-image P′. Suppose that the line l passes through the origin O and makes an angle α with the x-axis. If P has polar coordinates (r,θ), its mirror-image P′ has polar coordinates (r,2α − θ). In terms of Cartesian coordinates, reflection in the line l maps P with coordinates (x,y) to P′ with coordinates (x′,y′), where

In 3-dimensional space, an orthogonal matrix with determinant –1 and trace 1 represents a reflection in a plane containing the origin.

In higher dimensions, in Rn, reflection in the hyperplane with equation r∙n = c is given by

If c = 0, this is represented by an orthogonal matrix. In general, reflections are isometries.