||Unrestricted control polyhedra facilitate modeling free-form surfaces of arbitrary topology and local patch-layout by allowing га-sided, possibly nonplanar, facets and m-valent vertices. By cutting off edges and corners, the smoothing of an unrestricted control polyhedron can be reduced to the smoothing of a planar-cut polyhedron. A planar-cut polyhedron is a generalization of the well-known tensor-product control structure. The routine Pcp2Nurb in turn translates planar-cut polyhedra to a collection of four-sided linearly trimmed bicubic B-splines and untrimmed biquadratic B-splines. The routine can thus serve as central building block for overcoming topological constraints in the mathematical modeling of smooth surfaces that are stored, transmitted, and rendered using only the standard representation in industry. Specifically, on input of a nine-point subnet of a planar-cut polyhedron, the routine outputs a trimmed bicubic NURBS patch. If the subnet does not have geometrically redundant edges, this patch joins smoothly with patches from adjacent subnets as a four-sided piece of a regular C1 surface. The patch integrates smoothly with untrimmed biquadratic tensor-product surfaces derived from subnets with tensor-product structure. Sharp features can be retained in this representation by using geometrically redundant edges in the planar-cut polyhedron. The resulting surface follows the outlines of the planar-cut polyhedron in the manner traditional tensor-product splines follow the outline of their rectilinear control polyhedron. In particular, it stays in the local convex hull of the planar-cut polyhedron.
Categories and Subject Descriptors: D.3.2 [Programming Languages]: Language Classifications—C; G.l.l [Interpolation]: Spline and Piecewise Polynomial Interpolation; G.1.2 [Approximation]: Spline and Piecewise Polynomial Approximation; 1.3.5 [Computational Geometry and Object Modeling]: Boundary Representations; Surface Representations
General Terms: Algorithms
Additional Key Words and Phrases: Arbitrary patch layout, arbitrary surface topology, biquadratic tensor-product B-splines, C1 surface, free-form surface, Matlab, NURBS, planar-cut polyhedron, trimmed bicubic B-splines