Definition:Rotation (Geometry)/Space

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:

Then:

$\map {r_\theta} P = P'$

Thus $r_\theta$ is a rotation (in space) of (angle) $\theta$ about (the axis) $O$.

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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 $r_\theta$.

Vector Form

A space rotation $r_\theta$ can be expressed as an axial vector $\mathbf r_\theta$ such that:

the direction of $\mathbf r_\theta$ is defined to be its axis of rotation

the length of $\mathbf r_\theta$ specifies its angle of rotation of $\mathbf r_\theta$ to an appropriate scale.

Right-Hand Rule

Let $\mathbf V$ be an axial vector acting with respect to an axis of rotation $R$.

Consider a right hand with its fingers curled round $R$ so that the fingers are pointed in the direction of rotation of $\mathbf V$ around $R$.

The right-hand rule is the convention that the direction of $\mathbf V$ is the direction in which the thumb is pointing:

Also see

• Results about geometric rotations can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): rotation: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): rotation: 1.