Date of Completion

8-22-2013

Embargo Period

8-22-2013

Keywords

Bezier curve; subdivision; knot; homeomorphism; ambient isotopy; convergence; total curvature; computer graphics; visualization.

Major Advisor

Thomas J. Peters

Associate Advisor

Maria Gordina

Associate Advisor

William Abikoff

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

There is contemporary interest to preserve appropriate topological characteristics during geometric modeling. Here we focus upon topological equivalence (by home- omorphism) and isotopic equivalence (by ambient isotopy). Homeomorphism is an equivalence relation used for static images, while ambient isotopy requires a homeomorphism at each value of the time parameter, which is particularly applicable for time varying models. We provide sufficient conditions that guarantee these equivalences for geometric approximations of Bezier curves, as one of the fundamental computational representations. We also present further generalization beyond any curve approximation algorithm to establish broad criteria for a sequence of piecewise linear curves to become ambient isotopic to a given smooth curve. The criteria rely upon distance and total curvature, with upper bounds provided. Apart from the major theorems and algorithms, we investigate topological differences based on knot visualization and numerical analysis.

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