Understanding Quaternions

In Computer Graphics quaternions have three principal applications: to increase speed and reduce memory for calculations involving rotations; to avoid distortions arising from numerical inaccuracies caused by floating point computations with rotations; and to interpolate between two rotations for key frame animation.

Yet while the formal algebra of quaternions -- multiplication, sandwiching, interpolation -- is well established in Computer Graphics, the geometry of quaternions is not well understood. The purpose of this talk is to develop a better intuitive understanding of the geometry of quaternions, and to remove much of the mystery surrounding quaternion multiplication.

The main goals of this talk are to make four principal contributions:
1. To provide a fresh, geometric interpretation for quaternions, appropriate to contemporary Computer Graphics;
2. To present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in 3-dimensions;
3. To develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection;
4. To show how to apply sandwiching with quaternions to compute perspective projections.