Analytic functions in computer-aided geometric design

Abstract

This thesis presents a new paradigm for geometric modeling based on analytic functions. This model includes not only a representation of analytic curves and surfaces, but also tools and algorithms to manipulate this representation. Analytic functions on a given domain represent a very large class of infinitely smooth functions, including trigonometric functions and functions with poles outside the domain. Thus, the model is very rich; in particular the model is able to represent an object of optimal smoothness as well as functions as close as desired to singularities. The Bezier representation for polynomials generalizes to the Poisson representation for analytic curves and surfaces. The coefficients in the Poisson basis not only characterize an analytic function, but also are geometrically meaningful and intuitive control parameters for the curve or surface the function defines---as the Bezier control points are for polynomial shapes. Based on this Poisson representation, we derive standard geometric modeling algorithms for analytic curves and surfaces, including subdivision, trimming, evaluation and change of basis procedures. We also develop a notion of blossoming, as well as a de Boor-Fix formula and a Marsden identity, for analytic curves. These algorithms and tools provide an efficient and complete framework for using analytic functions in geometric modeling