Cones on Bezier curves and surfaces.
Deok‐Soo Kim, Tony C. Woo, Panos Y. Papalambros · Deep Blue (University of Michigan) · 1990
This thesis presents the definitions, operations, and applications of three types of cones for parametric curves and surfaces. A tangent cone for a parametric surface is a set of points in ${\cal E}\sp3$ such that the position vector of a point in the cone corresponds to a tangent vector on the surface, at some parameter values u and v. A normal cone for a parametric surface is a set of points in ${\cal E}\sp3$ such that the position vector of a point in the cone corresponds to a normal vector on the surface, at some parameter values u and v. A visibility cone for a parametric surface is defined as a set of points in ${\cal E}\sp3$ such that any line parallel to the position vector of a point in the visibility cone intersects with the surface at most once for all parameter values u and v. This thesis discusses the properties of the cones, and the relationships among the distinct cones. Computations of cones are discussed for Bezier representation of curves and surfaces. The hodograph, which is an instance of a first derivative surface of a Bezier surface, is used to compute a tangent cone. Since a hodograph is also another Bezier surface, the control points of a hodograph bound all tangent vectors of a given surface. The detailed analyses for the conditions under which the control points of a hodograph can be safely used to estimate the tangent cones are provided. Using the properties of the cones, a normal cone can be computed from tangent cones, and a visibility cone can be computed from a normal cone. Using the projection invariant property of the cones, the set operations such as union, intersection, and difference between cones are transformed to the polygon set operations.