Coordinate rotations and reflections

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 to its image on the other side of line . Then reflect to its image on the other side of line . If lines and make an angle with one another, then points and will make an angle around point , the intersection of and . I.e., angle will measure .

A pair of rotations about the same point will be equivalent to another rotation about point . 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 by an angle be denoted as . Let a reflection about a line through the origin which makes an angle with the -axis be denoted as . 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), \\ pt \operatorname(\theta) \, \operatorname(\phi) &= \operatorname(2\theta - 2\phi), \\ pt \operatorname(\theta) \, \operatorname(\phi) &= \operatorname(\phi + \tfrac\theta), \\ pt \operatorname(\phi) \, \operatorname(\theta) &= \operatorname(\phi - \tfrac\theta).
\end

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: . Every rotation has an inverse . Every reflection is its own inverse. Composition has closure and is associative, since matrix multiplication is associative.

Notice that both and 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: .

The following table gives examples of rotation and reflection matrix :

Rotation of axes

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

References

Sources

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{{DEFAULTSORT:Coordinate Rotations And Reflections
Euclidean symmetries
Euclidean plane geometry
Rotation