Quaternions in 3d computer graphics

Introduction

This short note is about how and why quaternions are sometimes used in 3d computer graphics. Sadly, the "why" doesn't seem to be documented anywhere that I can find.

In short, the main reason is that it becomes computationally easy to interpolate between two 3d rotations. This is often needed to smoothly animate between two orientations of a rigid object. Here are some of the details:

Linear Interpolation

A 3d rotation can be represented by a 3 by 3 square matrix M that operates on a column 3 vector v = (a,b,c)^T via matrix multiplication. If there are two rotations M₀ and M₁, then the linear interpolation M(t)=M₀(1-t)+M₁t will typically lead to matrices that are not rotations. If you stay in the world of 3 by 3 matrices, the standard way to fix this kind of thing is by Gram-Schmidt orthogonalization. Some people aren't happy with that (two reasons that come to mind are computational efficiency, and weird artifacts created by the fact that the path in the space of rotations isn't well controlled). This is where quaternions come in.

Firstly, represent the 3 vector as the "pure" quaternion p = ai+bj+ck. Secondly, a quaternion q defines a rotation via qp(1/q).Now linear interpolation q(t) = q₀(1-t)+q₁t just works. No post processing to correct the results is needed.

What we have here is a mapping from the invertible quaternions to the 3d rotations. It's clearly not one to one, since for any non-zero real number r, both q and rq result in the same rotation. However, the map does define an isomorphism from RP³=(R⁴-{0}/R-{0}) to SO(3). If you are familiar with these kinds of spaces, you probably know that RP³ is double covered by the simply connected space S³ (just like the projective plane RP² is double covered by the two sphere S²), and that SO(3) is double covered by the simply connected SU(2). As a consistency check, it's well known that S³ and SU(2) are isomorphic.

Feedback

If you have corrections, additions, modifications, etc please let me know mailto:walterv@gbbservices.com