Rotation Matrices and Homogeneous Transformations

A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n = 2, a coordinate frame is defined by an X axis and a Y axis with these two axes being orthogonal to each other. The intersection between the X axis and the Y axis is the origin of the coordinate frame. For a three-dimensional (3D) space, i.e., n = 3, a coordinate frame is defined by an X axis, a Y axis, and a Z axis, with each pair of these axes being orthogonal to each other. The X, Y , and Z axes meet at a point called the origin of the coordinate frame. Also, we will typically consider only right-handed coordinate frames, i.e., frames wherein the cross-product of the X axis unit vector and the Y axis unit vector is a unit vector in the Z axis direction. Given two coordinate frames (in either 2D or 3D spaces), there can, in general, be a translation (offset between the origins of the coordinate frames) and a rotation between the coordinate frames.