Definition:Rotation (Geometry)
This page is about Rotation in the context of Geometry. For other uses, see Rotation.
Definition
A rotation in the context of Euclidean geometry is an isometry from a Euclidean Space $\R^n$ as follows.
A rotation is defined usually for either:
- $n = 2$, representing the plane
or:
- $n = 3$, representing ordinary space.
Rotation in the Plane
A rotation $r_\alpha$ in the plane is an isometry on the Euclidean Space $\Gamma = \R^2$ as follows.
Let $O$ be a distinguished point in $\Gamma$, which has the property that:
- $\map {r_\alpha} O = O$
That is, $O$ maps to itself.
Let $P \in \Gamma$ such that $P \ne O$.
Let $OP$ be joined by a straight line.
Let a straight line $OP'$ be constructed such that:
- $(1): \quad OP' = OP$
- $(2): \angle POP' = \alpha$ such that $OP \to OP'$ is in the anticlockwise direction:
Then:
- $\map {r_\alpha} P = P'$
Thus $r_\alpha$ is a rotation (in the plane) of (angle) $\alpha$ about (the axis) $O$.
Rotation in Space
A rotation $r_\theta$ in space is an isometry on the Euclidean Space $\Gamma = \R^3$ as follows.
Let $AB$ be a distinguished straight line in $\Gamma$, which has the property that:
- $\forall P \in AB: \map {r_\theta} P = P$
That is, all points on $AB$ map to themselves.
Let $P \in \Gamma$ such that $P \notin AB$.
Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$.
Let a straight line $OP'$ be constructed perpendicular to $AB$ such that:
- $(1): \quad OP' = OP$
- $(2): \quad \angle POP' = \theta$ such that $OP \to OP'$ is in the anticlockwise direction:
Then:
- $\map {r_\theta} P = P'$
Thus $r_\theta$ is a rotation (in space) of (angle) $\theta$ about (the axis) $O$.
Axis of Rotation
Let $r_\theta$ be a rotation in the Euclidean Space $\Gamma = \R^n$.
The set $A$ of points in $\Gamma$ such that:
- $\forall P \in A: \map {r_\theta} P = P$
is called the axis of rotation of $r_\theta$.
Angle of Rotation
Let $r_\theta$ be a rotation in the Euclidean Space $\Gamma = \R^n$.
Let $\map {r_\theta} P = P'$ about a point $O$ on the axis of rotation of $r_\theta$.
- The number $\theta$ which defines the angle $POP'$ is called the rotation angle or angle of rotation of $r_\theta$.
Also see
- Results about geometric rotations can be found here.
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(5)$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): rotation