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 P {\displaystyle P} to its image P ′ {\displaystyle P'} on the other side of line L 1 {\displaystyle L_{1}} . Then reflect P ′ {\displaystyle P'} to its image P ″ {\displaystyle P''} on the other side of line L 2 {\displaystyle L_{2}} . If lines L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} make an angle θ {\displaystyle \theta } with one another, then points P {\displaystyle P} and P ″ {\displaystyle P''} will make an angle 2 θ {\displaystyle 2\theta } around point O {\displaystyle O} , the intersection of L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} . I.e., angle ∠ P O P ″ {\displaystyle \angle POP''} will measure 2 θ {\displaystyle 2\theta } . A pair of rotations about the same point O {\displaystyle O} will be equivalent to another rotation about point O {\displaystyle 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.
Mathematical expression The statements above can be expressed more mathematically. Let a rotation about the origin O {\displaystyle O} by an angle θ {\displaystyle \theta } be denoted as Rot ( θ ) {\displaystyle \operatorname {Rot} (\theta )} . Let a reflection about a line L {\displaystyle L} through the origin which makes an angle θ {\displaystyle \theta } with the x {\displaystyle x} -axis be denoted as Ref ( θ ) {\displaystyle \operatorname {Ref} (\theta )} . 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,
Rot ( θ ) = [ cos θ − sin θ sin θ cos θ ] , {\displaystyle \operatorname {Rot} (\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}},}
and likewise for a reflection,
Ref ( θ ) = [ cos 2 θ sin 2 θ sin 2 θ − cos 2 θ ] . {\displaystyle \operatorname {Ref} (\theta )={\begin{bmatrix}\cos 2\theta &\sin 2\theta \\\sin 2\theta &-\cos 2\theta \end{bmatrix}}.}
With these definitions of coordinate rotation and reflection, the following four identities hold:
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