2. Rotations as transformations

The transformation $T($x$) =$   x$P$ that comes from a rotation matrix $P$ is a rotation of a special kind: Because it is a homogeneous linear transformation, it must leave the origin fixed. Thus in R$^2$, a rotation matrix gives a rotation about the origin (a rotation whose center is the origin); in R$^3$, a rotation matrix gives a rotation whose axis goes through the origin.

In graphics, however, it is often necessary to rotate in R$^2$ about other centers and in R$^3$ about axes not through the origin. Precisely how to make such rotations will be discussed as part of the study of ``affine transformations''. For now, just consider this definition:

Definition 2.1 . A transformation R$^n \rightarrow$   R$^n$ is a rotation if it is rigid, preserves orientation, and leaves at least one point fixed (i.e., takes the point to itself).

If a rotation leaves the origin fixed, then it is a homogeneous linear transformation coming from a rotation matrix. (This fact is not hard to prove.)

Agreement. In talking about rotations, let's assume we are talking about rotations that leave the origin fixed, unless it is obvious that we are not.