Ross Richardson

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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The Senior Thesis Program Department of Mathematics Harvey Mudd College	Ross Richardson Harvey Mudd College Mathematics 2003 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. Download: PDF

Ross Richardson

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

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