Rotations and reflections in two dimensions

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

Process

A rotation in the plane can be formed by composing a pair of reflections. First reflect a point Template:Mvar to its image Template:Mvar on the other side of line L₁. Then reflect Template:Mvar to its image Template:Mvar on the other side of line L₂. If lines L₁ and L₂ make an angle Template:Mvar with one another, then points Template:Mvar and Template:Mvar will make an angle 2θ around point Template:Mvar, the intersection of L₁ and L₂. I.e., angle ∠ POP′′ will measure 2θ.

A pair of rotations about the same point Template:Mvar will be equivalent to another rotation about point Template:Mvar. On the other hand, the composition of a reflection and a rotation, or of a rotation and a reflection (composition is not commutative), will be equivalent to a reflection.

Mathematical expression

The statements above can be expressed more mathematically. Let a rotation about the origin Template:Mvar by an angle Template:Mvar be denoted as Rot(θ). Let a reflection about a line Template:Mvar through the origin which makes an angle Template:Mvar with the Template:Mvar-axis be denoted as Ref(θ). Let these rotations and reflections operate on all points on the plane, and let these points be represented by position vectors. Then a rotation can be represented as a matrix, $$\mathrm{Rot}(\theta )=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right],$$

and likewise for a reflection, $$\mathrm{Ref}(\theta )=\left[\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right].$$

With these definitions of coordinate rotation and reflection, the following four identities hold: $$\begin{array}{rl}\mathrm{Rot}(\theta )\mathrm{Rot}(\varphi )& =\mathrm{Rot}(\theta +\varphi ),\\ \mathrm{Ref}(\theta )\mathrm{Ref}(\varphi )& =\mathrm{Rot}(2\theta -2\varphi ),\\ \mathrm{Rot}(\theta )\mathrm{Ref}(\varphi )& =\mathrm{Ref}(\varphi +\frac{1}{2}\theta ),\\ \mathrm{Ref}(\varphi )\mathrm{Rot}(\theta )& =\mathrm{Ref}(\varphi -\frac{1}{2}\theta ).\end{array}$$

Proof

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. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is associative, since matrix multiplication is associative.

Notice that both Ref(θ) and Rot(θ) have been represented with orthogonal matrices. These matrices all have a determinant whose absolute value is unity. Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1.

The set of all orthogonal two-dimensional matrices together with matrix multiplication form the orthogonal group: O(2).

The following table gives examples of rotation and reflection matrix :

Type	angle θ	matrix
Rotation	0°	$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
Rotation	$\pm$45°	$\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& \mp 1\\ \pm 1& 1\end{array}\right)$
Rotation	90°	$\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$
Rotation	180°	$\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$
Reflection	0°	$\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$
Reflection	45°	$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$
Reflection	90°	$\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$
Reflection	-45°	$\left(\begin{array}{cc}0& -1\\ -1& 0\end{array}\right)$

See also

• 2D computer graphics#Rotation
• Cartan–Dieudonné theorem
• Clockwise
• Dihedral group
• Euclidean plane isometry
• Euclidean symmetries
• Instant centre of rotation
• Orthogonal group
• Rotation group SO(3) – 3 dimensions