Rotation Matrices and Formulas
Here is information about using matrices to do three-dimensional rotations about an arbitrary axis.
The paper as a PDF.
The paper as a web page: Matrix Rotations About an Arbitrary Axis in 3 Dimensions.
Java code for the matrices: see the zip file in the attachments below.
See the rotations in action at http://twist-and-shout.appspot.com/.
This may be of interest to those looking for some concrete formulas for matrix rotations. Since the original posting (September 7, 2005), this has been a popular reference for people seeking complete formulas for rotations about a line not through the origin. Similar expositions for the simpler problem of rotations about an arbitrary axis through the origin are found in the Nested dimensions section of the Rotation matrix article on Wikipedia (the matrix Q_u(theta) in the Wikipedia article is the matrix for rotation about the origin I give in Rotation About an Arbitrary Axis in 3 Dimensions). The last time I checked these articles lacked formulas for rotations about arbitrary axes or rotations as functions. Those interested in another way to do rotations are encouraged to investigate rotations with quaternions.
Below I've attached Java code for the rotations described in the paper. The code uses the same notations as in the paper.
Demo
As a programming exercise I've written a web application that gives a visualization of rotations at http://twist-and-shout.appspot.com/.
Interactive demo at http://twist-and-shout.appspot.com/.