An introduction to solid modeling
An introduction to solid modeling
Generalized B-spline surfaces of arbitrary topology
SIGGRAPH '90 Proceedings of the 17th annual conference on Computer graphics and interactive techniques
An implementation of triangular B-spline surfaces over arbitrary triangulations
Selected papers of the international symposium on Free-form curves and free-form surfaces
SIAM Journal on Numerical Analysis
ACM Transactions on Graphics (TOG)
Computing moments of objects enclosed by piecewise polynomial surfaces
ACM Transactions on Graphics (TOG)
Localized-hierarchy surface splines (LeSS)
I3D '99 Proceedings of the 1999 symposium on Interactive 3D graphics
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Unrestricted control polyhedra facilitate modeling free-form surfaces of arbitrary topology and local patch-layout by allowing n-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.