Definition:Rotation (Geometry)/Space

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Definition

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) $O$.