Definition:Rotation (Geometry)

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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:
Rotation-in-Plane.png


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:


Rotation-in-Space.png


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 angle of rotation of $r_\theta$.


Also see

  • Results about geometric rotations can be found here.


Sources