Surface Approximation By Low Degree Patches With Multiple Representations

Abstract

Computer Aided Geometric Design (CAGD) is concerned with the representation and approximation of curves and surfaces when these objects have to be processed by a computer. Parametric representations are very popular because they allow considerable flexibility for shaping and design. Implicit representations are convenient for determining whether a point is inside, outside or on the surface. These representations offer many complimentary advantages. Therefore, it is desirable to build geometric models with surfaces which have both parametric and implicit representations. Maintaining the degree of the surfaces low is important for practical reasons. Both the size of the surface representation, as well as the difficulties encountered in the algorithms, e.g. root finding algorithms, grow quickly with increasing degree. This thesis introduces low degree surfaces with both parametric and implicit representations and investigates their properties. A new method is described for creating quadratic triangular Bezier surface patches which lie on implicit quadric surfaces. Another method is described for creating biquadratic tensor product Bezier surface patches which lie on implicit cubic surfaces. The resulting surface patches satisfy all of the standard properties of parametric Bezier surfaces, including interpolation of the corners of the control polyhedron and the convex hull property. The second half of this work describes a scheme for filling n-sided holes and for approximating the resulting smooth surface consisting of high degree parametric Bezier surface patches by a continuous surface consisting of low degree patches with both parametric and implicit representations. A new technique is described for filling an n-sided hole smoothly using a single parametric surface patch with a geometrically intuitive compact representation. Next, a new degree reduction algorithm is applied to approximate high degree parametric Bezier surfaces by low degree Bezier surfaces. Finally, a variant of the least squares technique is used to approximate parametric Bezier surfaces of low degree by low degree surfaces with both parametric and implicit representations. The resulting surfaces have boundary continuity and approximation properties.