Taxonomy of interpolation constraints on recursive subdivision surfaces

Part of this work was done during a visit by A. Nasri to Cambridge University The scenario we consider is that a surface is defined by a collection of interpolation constraints, either of points or of curves, such curves being themselves defined in terms of B-spline control polygons. The interpolation points and the vertices of the curve control polygons are then joined into a network of the desired topological structure, possibly with additional vertices to help provide some geometric control in regions sparsely provided with hard constraints (Nasri and Sabin 2001). Some preprocessing may then be done on this network to produce a new network which can be interpreted by one of the standard recursive division constructions. Alternatively, the interpolations may be taken into the subdivision process dynamically. The result of this is a surface of the required topological structure which can be controlled qualitatively in some regions and by hard interpolation constraints in others. Although this procedure is not yet complete in all details, specific interpolation results fit into this framework. The bulk of this paper describes and classifies the known results and identifies remaining open questions. A recursive division surface, S, is basically defined by a duple, (P0, R), where P0 is an initial configuration and R is a refinement procedure. The configuration consists of a set of vertices, edges, and faces. This is often referred to as a polyhedron even though the faces need not have coplanar vertices. The refinement procedure is a set of rules applied to a configuration to generate another with more vertices, edges and smaller faces than the initial one. At each level, i, of the refinement process, the polyhedron Pi−1 is taken as input to the refinement R, which produces another polyhedron, Pi , which may itself be taken as input to the next refinement step and so on. If R satisfies some conditions (Doo and Sabin 1978; Reif 1995; Zorin and Schröder 2000), then at the limit the sequence of polyhedra Pi converges to a smooth surface. There are many alternative sets of rules, R. They may typically be generated by using the knot-insertion procedure of some B-spline tensor-product surface to insert knots at the mid-points of the existing knot intervals. This induces a set of rules to be applied in the regular case. These rules take the form of coefficients used to form the coordinates of the vertices of Pi , by taking linear combinations of the vertices of Pi−1. These rules are then extended by defining appropriate linear combination coefficients to cover the irregular situations of non-regular vertices or faces.