Algorithm 783: Pcp2Nurb - smooth free-form surfacing with linearly trimmed bicubic B-splines

Unrestricted control polyhedra facilitate modeling free-form surfaces of arbitrary topology and local patch-layout by allowing n-sided, possibly non-planar facets and m-valent vertices. By cutting oo 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 topo-logical constraints in the mathematical modeling of smooth surfaces that are stored, transmitted and rendered using only the standard representation in industry. Speciically, 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 C 1 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. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proot or direct commercial advantage and that copies show this notice on the rst page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior speciic permission and/or a fee. Permissions may be requested from Publications Fig. 1. (top) input polyhedron; (middle) planar-cut polyhedron; (bottom) NURBS surface. 1. INTRODUCTION Polyhedra can be smoothed into free-form surfaces using a variety of approaches such as rational blends, generalized subdivision or simplex splines (see e.g. 3], 1], 2]). A major criticism leveled at these techniques is that they are incompatible, i.e. cannot be represented exactly or eeciently in the …