In Euclidean geometry, a rotation about the origin is an orientation-preserving orthogonal linear map, represented by an element of the special orthogonal group. About another center , the map has form . In three dimensions a nonidentity rotation fixes an axis.
The fixed straight line of a nonidentity three-dimensional rotation. For a rotation matrix , its direction vector is an eigenvector with eigenvalue . A unit vector along it also specifies the positive sense of the angle through the right-hand convention.
A rotation through an angle of . In three dimensions, a half-turn about a straight line with unit vector fixes its parallel component and negates its orthogonal complement, giving the rotation matrix by orthogonal projection.
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In mathematics, rotation refers to a transformation that turns a shape or object around a fixed point called the center of rotation. The amount of rotation is usually measured in degrees or radians. ### Key Concepts: 1. **Center of Rotation**: This is the point around which the rotation occurs. For example, if you rotate a triangle around one of its vertices, that vertex would be the center of rotation.