A tension-compatible patch for shape-preserving surface interpolation

Several issues and techniques related to the problem of constructing shape-preserving interpolants for discrete gridded data are discussed. For choosing slopes and twists it is shown that the method of parabolic blending is intuitively appealing and easy to implement, and that it produces results consistent with standard notions of implied shape. It is shown by examples that standard bicubic patches fail to respect the shape implied by this augmented data, and two alternative patching schemes based on adaptive tension techniques, which remedy the problem, are provided. The constructions are general enough to encompass many different tensioning methods, including the classical tension splines of D.G. Schweikert (1966) and rational splines. Both schemes used univariate techniques along grid lines to produce a network of shape-preserving curves. One scheme then makes further use of tensioned interpolation to provide the extra information that allows use of a general-purpose Coons patch. The other scheme involves the use of a novel patch ´customized´ for blending the independently tensioned boundary curves. The latter method performs comparably with the former, at considerably reduced cost.<>