Understanding quaternions

Quaternion multiplication can be applied to rotate vectors in 3-dimensions. Therefore in Computer Graphics, quaternions are sometimes used in place of matrices to represent rotations in 3-dimensions. Yet while the formal algebra of quaternions is well-known in the Graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this paper are:i.To provide a fresh, geometric interpretation of quaternions, appropriate for contemporary Computer Graphics;ii.To derive the formula for quaternion multiplication from first principles;iii.To present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in 3-dimensions based on insights from the algebra and geometry of multiplication in the complex plane;iv.To develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection;v.To show how to apply sandwiching to compute perspective projections.In Part I of this paper, we investigate the algebra of quaternion multiplication and focus in particular on topics i and ii. In Part II we apply our insights from Part I to analyze the geometry of quaternion multiplication with special emphasis on topics iii, iv and v.