Geometric transformations:

A geometric transformation is any bisection of a set having some geometric structure to itself or another such set. Specifically, "A geometric transformation is a function whose domain and range are sets of points. Most often the domain and range of a geometric transformation are both R2 or both R3. Often geometric transformations are required to be 1-1 functions, so that they have inverses." The study of geometry may be approached via the study of these transformations.

 Coordinate transformation:

A Cartesian coordinate system allows position and direction in space to be represented in a very convenient manner. Unfortunately, such a coordinate system also introduces arbitrary elements into our analysis. After all, two independent observers might well choose coordinate systems with different origins, and different orientations of the coordinate axes. In general, a given vector a will have different sets of components in these two coordinate systems. However, the direction and magnitude of a are the same in both cases. Hence, the two sets of components must be related to one another in a very particular fashion.