Math 2.0

Geometry Definition

Rotation

Builds on Euclidean isometryEuclidean plane

Definition

Let cc be a point of ℝn\mathbb{R}^n, and let AA be an orthogonal n×nn \times n matrix with det⁡A=1\det A = 1. The rotation about cc with the matrix AA is the map R ⁣:ℝn→ℝn,R(p)=c+A (p−c).R \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad R(p) = c + A\,(p - c). In the Euclidean plane ℝ2\mathbb{R}^2, the rotation about cc through the angle θ∈ℝ\theta \in \mathbb{R} is the case A=(cos⁡θ−sin⁡θsin⁡θcos⁡θ).A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.

Remarks

Each rotation is a direct Euclidean isometry that fixes cc. In the plane, each orthogonal matrix with determinant 11 has this form, so each plane rotation has an angle. In the plane, rotations about one centre commute, and their angles add modulo 2π2\pi. Rotations about different centres do not commute in general. In the plane, the composite of the reflections in two lines that meet at cc is a rotation about cc. Mathlib's Orientation.rotation is the rotation through an angle of an oriented Euclidean plane, as a linear map.

Reading path

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Euclidean isometry, concept, no evidence levelEuclidean isometryconceptEuclidean plane, object, no evidence levelEuclidean planeobjectRotation, definition, no evidence levelRotationdefinition
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  1. Euclidean isometry concept
  2. Euclidean plane object
  3. Rotation definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclidean_plane_isometry, Rotations
  2. Web https://en.wikipedia.org/wiki/Rotation_(mathematics), In Euclidean geometry
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DefinesRotation
UsesEuclidean isometry, Euclidean plane

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