In geometry, two-dimensional

rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

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 ''L''₁. Then reflect ''P''′ to its image ''P''′′ on the other side of line ''L₂''. If lines ''L''₁ and ''L''₂ make an angle ''θ'' with one another, then points ''P'' and ''P''′′ will make an angle ''2θ'' around point ''O'', the intersection of ''L''₁ and ''L₂''. I.e., angle ''POP′′'' will measure 2''θ''. A pair of rotations about the same point ''O'' will be equivalent to another rotation about point ''O''. 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. The statements above can be expressed more mathematically. Let a rotation about the origin ''O'' by an angle ''θ'' be denoted as Rot(''θ''). Let a reflection about a line ''L'' through the origin which makes an angle ''θ'' with the ''x''-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, :

\operatorname(\theta) = \begin
  \cos \theta & -\sin \theta \\
  \sin \theta &  \cos \theta
\end,

and likewise for a reflection, :

\operatorname(\theta) = \begin
  \cos 2 \theta &  \sin 2 \theta \\
  \sin 2 \theta & -\cos 2 \theta
\end.

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

\begin
  \operatorname(\theta) \, \operatorname(\phi) &= \operatorname(\theta + \phi), \\
  \operatorname(\theta) \, \operatorname(\phi) &= \operatorname(2\theta - 2\phi), \\
  \operatorname(\theta) \, \operatorname(\phi) &= \operatorname\left(\phi + \frac\theta\right), \\
  \operatorname(\phi) \, \operatorname(\theta) &= \operatorname\left(\phi - \frac\theta\right).
\end

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 In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...

. These matrices all have a determinant whose

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

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 In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

: ''O''(2). The following table gives examples of rotation and reflection matrix :

See also