Rotations and reflections in two dimensions

In Euclidean geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.

A rotation in the plane can be formed by composing a pair of reflections.

First reflect a point P to its image P′ on the other side of line L1.

Then reflect P′ to its image P′′ on the other side of line L2.

If lines L1 and L2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the intersection of L1 and L2.

Let a reflection about a line L through the origin which makes an angle θ with the x-axis be denoted as Ref(θ).

With these definitions of coordinate rotation and reflection, the following four identities hold:

These equations can be proved through straightforward matrix multiplication and application of trigonometric identities, specifically the sum and difference identities.

The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group.

Every reflection Ref(θ) is its own inverse.

These matrices all have a determinant whose absolute value is unity.

The following table gives examples of rotation and reflection matrix :