Geometric modeling is a relatively new discipline that focuses on the problem of constructing and manipulating representations of geometric objects. One of the fundamental problems is choosing an appropriate representation for surfaces. In one popular approach, surfaces are represented as zero sets of polynomials. Such surfaces, referred to as algebraic surfaces, include almost all of the surfaces currently used in geometric modeling (e.g., Bezier surfaces, B-spline surfaces, quadric surfaces). A benefit of the use of these surfaces is the existence of an extensive mathematical theory of algebraic surfaces known as algebraic geometry. We propose to apply theory from algebraic geometry to three current problems in geometric modeling: 1. Symbolic manipulation of systems of polynomial equations: We propose to investigate symbolic manipulation methods involving elimination theory (e.g., resultants). Specifically, we intend to explore new methods for eliminating several variables from a system of polynomials simultaneously. 2. Computing the intersection of algebraic surfaces: we propose to apply the theory of birational maps to the problem of constructing an algorithm for computing the intersection of algebraic surfaces. 3. Construction of piecewise smooth algebraic surfaces: we propose to use the theory of polynomial ideals to aid in the design and construction of a system of piecewise continuous algebraic surfaces. In particular, we hope to develop methods for constructing free-form algebraic surfaces.
