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How to do describe a general body translation and rotation?

How do you describe a rotation and translation of a rigid body in 3 dimensional space? Say the "center" of the body has location (xo,yo,zo) with respect to the origin. Also say vector (x1,y1,z1) extends from the center of the body and is fixed with respect to the body.

Public Comments

1. The simplest way is to give the velocity of the center of rotation, and the angular velocity around 3 axises of rotation (x,y,z for example with the center of rotation as the origin).
2. The best and most elegant way to handle translation and rotations is through matrices. A single matrix can represent rotation, scaling, translation, skewing, coordinate inversions, projections... it is very powerful! If you aren't used to matrix mathematics, it can take some getting used to. But once you do, a lot of bookkeeping disappears. These sorts of matrix transformations are used all the time in physics and 3D graphics. I've got a link here to one example (note how the author uses 4x4 matrices to include translations naturally) but some web searches will quickly show other sources.