Definition:Rotation (Geometry)/Plane
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Definition
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) $O$.
Also see
- Results about geometric rotations can be found here.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 21$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): rotation: 2. (in the plane)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): rotation: 2. (in the plane)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): rotation (of the plane)