### Monday April 22, 2002

**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.