Geometric transformation

In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale).

More specifically, it is a function whose domain and range are sets of points – most often both

Starting with a row vector v, the right action of the composed transformation is w = vAB.

Geometric transformations can be distinguished into two types: active or alibi transformations which change the physical position of a set of points relative to a fixed frame of reference or coordinate system (alibi meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the frame of reference or coordinate system relative to which they are described (alias meaning "going under a different name").

[citation needed] For instance, active transformations are useful to describe successive positions of a rigid body.

On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the tibia relative to the femur, that is, its motion relative to a (local) coordinate system which moves together with the femur, rather than a (global) coordinate system which is fixed to the floor.

[11] In three-dimensional Euclidean space, any proper rigid transformation, whether active or passive, can be represented as a screw displacement, the composition of a translation along an axis and a rotation about that axis.

In the active transformation (left), a point P is transformed to point P ′ by rotating clockwise by angle θ about the origin of a fixed coordinate system. In the passive transformation (right), point P stays fixed, while the coordinate system rotates counterclockwise by an angle θ about its origin. The coordinates of P ′ after the active transformation relative to the original coordinate system are the same as the coordinates of P relative to the rotated coordinate system.