MAT 351

MAT 551

Quaternions, Interpolation and Animation (3 Cr.)

Prerequisite: MAT 300

Description: This course covers topics in abstract algebra and geometry woven together by the thread of quaternions. This particular thread is chosen with an eye toward applications in computer graphics, specifically to the interpolation of rotation operators and their relations to animation. The course will also present a self-contained summary of abstract algebra, including elements of finite groups, rings, fields, and real algebras. Students will pay attention to certain examples of these structures, including the complex numbers as an algebra, and the unit complex numbers as the rotation group in the plane. These examples in two dimensions are then extended to three and four dimensions with the study of division algebras and Hamilton’s quaternion algebra. In particular, the unit quaternion sphere is exhibited as a two-fold cover of the rotation group of threespace. The second part of the course will review basic differential geometry, continuity, and differentiability of curves in four-space. Here the students will have the opportunity to combine their knowledge of abstract algebra with their knowledge of parametrized curves and produce some interesting visual applications to three-dimensional animation.