Speaker: Richard Palais
Title: Some Reflections on Rotations (Rapid Rotation of Objects in 3D Graphics)
Abstract: What is the "best" (i.e., quickest) way to rotate 3D objects on a computer screen? Since rotation (together with translation, which is trivial) is the basis for 3D animation, this problem is crucial for the efficient implementation of many 3D computer graphic rendering algorithms. Yet, despite its apparent simplicity of statement, this turns out to be a surprisingly complex and interesting problem, not only for practical computer science, but also in numerical analysis and pure mathematics.
There are many ways to represent rotations---three by three orthogonal matrices, elements of SU(2) (i.e., 2 by 2 unitary matrices of determinant 1), unit quaternions, spinors, Euler angles, elements of projective three space---and we shall discuss each of them (and introduce a new one!). There has been a lot of debate in recent years about the relative merits of each---but of course our new one is the best :-). In fact what we shall see is that the best representation of a rotation and the best way to rotate an object are closely related questions, and both of them are highly dependent on the way that the data that defines the rotation is presented, and on the nature of the object that is to be rotated.