Quadric surfaces such as cylinders and spheres play an important role in CAD. This paper describes a new method for creating triangular surface patches on a quadric surface. The surface patches are defined using a restricted type of quadratic B~zier control polyhedron. The control polyhedron and the resulting quadric surface patch satisfy all of the standard properties of parametric B6zier surfaces, including interpolation of the corners of the control polyhedron and the convex hull property. A new technique for creating a C 7 mesh of these quadric surface patches is also introduced.
quadric surfaces, B~zier representation, triangular patches, control polyhedron
Simple curved surfaces such as spheres, cylinders and hyperboloids play a fundamental role in computeraided geometric design. Each is a quadric surface and can thus be defined as the zero contour of a quadratic function. Quadric surfaces and the volumes they enclose are often the primitive elements in CSG representations 1, but one difficulty has complicated the use of quadric surfaces in geometric modelling: the lack of a convenient method for creating bounded patches on a quadric surface.
A parametric surface, representation such as the Bernstein-B~zier representation allows a control polyhedron to specify a bounded portion of a surface. Given a control polyhedron P, B&zier methods generate a surface patch S that approximates P. Several desirable properties relate S and P. For example, S interpolates the corners of P, is tangent to P at its corners, and lies in the convex hull of P. A method for creating quadric surface patches and associated control polyhedra that are similarly related would greatly simplify the process of modelling with quadrics.
Several methods have been proposed to address this problem for particularly important types of quadrics like spheres and cylinders 2-7. A method based on general quadratic Bezier surfaces is unlikely because, in general, a triangular parametric quadratic B~zier surface patch has an implicit degree of four ~. Under certain restrictions, however, such quadratic para-