Taking a twist is understood in the geometry of a self-map of Euclidean space with at least one fixed point, which leaves all distances invariant and preserves the orientation. If the orientation is reversed, there is a mirroring or rotation-reflection.
Since rotations lengths (and hence angle) can be invariant, every turn is a congruence.
In the Euclidean plane, any rotation can be precisely determined a point, the center of rotation. If a different point of the plane and its image, the angle does not depend on and defines the rotation angle. A 180 ° rotation causes the same mapping of the plane as a point reflection on the turning center.
In three-dimensional space, any rotation can be an axis of rotation fixed pointwise. , Each plane perpendicular to this axis, a fixed point (namely, the intersection with the axis) and is represented by the rotation of itself.
In analytic geometry rotations are length-preserving affine transformations. If we choose a Cartesian coordinate system whose origin is on the axis of rotation, then the translational zero share. The rotation is then described by a rotation matrix, which is a orthogonal matrix whose determinant is 1.