Why are quaternions used for rotations?

Question

I\'m a physicist, and have been learning some programming, and have come across a lot of people using quaternions for rotations instead of writing things in matrix/vector fo

Answer 1

In physics, there are very good reasons we don't use quaternions (despite the bizarre story that's occasionally told about Hamilton/Gibbs/etc). Physics requires that our descriptions have good analytic behavior (this has a precisely defined meaning, but in some rather technical ways that go far beyond what's taught in normal intro classes, so I won't go into any detail). It turns out that quaternions don't have this nice behavior, and so they aren't useful, and vectors/matrices do, so we use them.

Well, I am a physicist, too. And there are some situations where quaternions simply rock! Spherical Harmonics for example. You have two atoms scattering, exchanging an electron: what is the orbital spin transfer? With quaternions it is just multiplication i.e. summing up the exponents of the SH base functions expressed as quaternions. (Getting the Legendre Polynomials into quaternion notation is a bit tedious though).

But I agree, they are not a universal tool, and especially in rigid body mechanics they would be very cumbersome to use. Yet to cite Bertrand Russell answer in question of a student how much math a physicist needs to know: "As much as possible!"

Anyway: Why do we love quaternions in computer graphics? Because they have a number of appealing properties. First one can nicely interpolate them, which is important if one is animating rotating things, like the limbs around a joint. With a quaternion it is just scalar multiplication and normalization. Expressing this with a matrix requires evaluation of sin and cos, then building a rotation matrix. Then multiplying a vector with a quaternion is still cheaper as going through a full vector-matrix multiplication, it is also still cheaper if one adds a translation afterwards. If you consider a skeletal animation system for a human character, where one must evaluate a lot of translation/rotations for a large number of vertices, this has a huge impact.

Another nice side effect of using quaternions is, that any transformation inherently is orthonormal. With translation matrices one must re-orthonormalize every couple of animation steps, due to numerical round-off errors.