The Senior Thesis Program

Department of Mathematics
Harvey Mudd College

Ross Richardson

Harvey Mudd College Mathematics 2003

A strange picture of me
Primary research interests: Combinatorics, Graph Theory, Computational Geometry, Theoretical Computer Science

Other mathematical interests: Differential Geometry, Geometric Topology

Starting in the fall of 2003, I will be a graduate student in the Department of Mathematics at the University of California, San Diego in beautiful La Jolla.

Senior Thesis

Designing Smooth Motions of Rigid Objects:
Computing Curves in Lie Groups

Advisor: Weiqing Gu
Second Reader: Zachary Dodds (Computer Science)

Abstract

Consider the problem of designing the path of a camera in 3D. As we may identify each camera position with a member of the Euclidean motions, SE(3), the problem may be recast mathematically as constructing interpolating curves on the (non-Euclidean) space SE(3).

There exist many ways to formulate this problem, and indeed many solutions. In this thesis we shall examine solutions based on simple geometric constructions, with the goal of discovering well behaved and computable solutions. In affine spaces there exist elegant solutions to the problem of curve design, which are collectively known as the techniques of Computer Aided Geometric Design (CAGD). The approach of this thesis will be the generalization of these methods and an examination of computation on matrix Lie groups. In particular, the Lie groups SO(3) and SE(3) will be examined in some detail.

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