Smooth mesh interpolation with cubic patches

An algorithm for the local interpolation of a mesh of cubic curves with 3-and 4-sided facets by a piecewise cubic C ~ surface is stated and illustrated by an implementation. Precise necessary and sufficient conditions for oriented tangent-plane continuity between adjacent patches are derived, and the explicit constructions are characterized by the degree of the three scalar weight functions that relate the versal to the two transversal derivatives. The algorithm fully exploits the possibility of reparametrization by choosing all three weight functions nonconstant and not just degree-raising polynomials. The construction is local and consists mainly of averaging. The only systems to be solved are linear and of size 2 x 2. The algorithm guarantees interpolating surfaces without cusps and has a simple, implemented extension to n-sided facets. This paper presents an algorithm that constructs a smooth piecewise polynomial surface interpolant to a mesh of cubic curves by splitting and averaging. The algorithm, called SPLAV in the following, is similar to the schemes described by Farin 1, Piper 2, Shirman and S~quin 3, and Jones 4. It differs in that • the surface interpolant is of lower degree, • the surface interpolant can be guaranteed to be free of cusps, • no systems of equations (larger than 2 x 2) have to be solved, • the construction extends to arbitrary n-sided mesh facets. Table I, taken from Peters ~, gives the cletails. SPLAY is a splitting_scheme. Compared to blending schemes (e.g. Nielson 6, Charrot and Gregory7), splitting schemes use the same number of pieces to cover an n-facet, but generate generically a lower-degree surface. A problem with cubic mesh interpolation by cubics was pointed out" in Piper 2. This paper characterizes admissible data for cubic mesh interpolation with splitting and exhibits a scheme for generating admissible curve meshes from data points (and their normals). Data Sides Degree Reference N 3 4 Farin ~ T 3 4 Piper 2 M 3, 4 4 Shirman and S~quin 3 N n 5 Jones 4 M 3,4 3 Peters 8 and this paper M n > 4 4 this paper N 3 bi3 Peters 9 N = normal, T = tangents, M = cubic mesh SPLAV is summarized as follows. Each n-sided mesh facet splits into n subtriangles. If n-3 or n = 4 or if the data are symmetric, then a piecewise cubic interpolant (in Bernstein-B&zier form) is constructed; otherwise the interpolant is quartic. …