On surface design with implicit algebraic surfaces

Insung Ihm, Purdue University

Abstract

Computer Aided Geometric Design (CAGD) is a rapidly growing area that involves theories and techniques from many disciplines such as computer science and mathematics as well as engineering. One of the most important subjects in CAGD is to efficiently model physical objects with a surface or collection of surfaces for many applications of CAD/CAM, computer graphics, medical imaging, robotics and etc. Most research in surface modeling has been largely dominated by the theory of parametrically represented surfaces. While they have been successfully used in representing physical objects, parametric surfaces are confronted with some problems when objects represented with them are manipulated in geometric modeling systems. In recent years, increasing attention has been paid to algebraic surfaces that are implicitly defined by a polynomial equation, and provide a more general class of surfaces at lower degrees. In this thesis, we consider the problem of modeling complex geometric objects with smooth piecewise algebraic surface patches. We present an interpolation algorithm, called Hermite interpolation, which characterizes a class of all algebraic surfaces of a specified degree that interpolate given points and space curves with tangent plane continuity. The Hermite interpolation algorithm with least squares approximation transforms the geometric problem of algebraic surface design into a linear algebra problem which can be solved efficiently. Based on this algebraic model, we explore the class of quintic algebraic surfaces to smooth convex polyhedra with a mesh of smooth piecewise algebraic surface patches. Degrees of freedom in constructing wire frames for polyhedra are used to control shapes of curved models of polyhedra. The open problem of modeling polyhedra having arbitrary shapes with quintic triangular algebraic surface patches is considered. Finally, we present a heuristic algorithm which quickly computes a good piecewise linear approximation of a given digitized space curve. This algorithm serves as a primary tool in polygonizing triangular algebraic surface patches.

Degree

Ph.D.

Advisors

Bajaj, Purdue University.

Subject Area

Computer science

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