Multivariable Curve Interpolation

Abstract

The problem of defining a smooth surface through an array of points in space is well known. Several methods of solution have been proposed. Generally, these restrict the set of points to be one-to-one defined over a planar rectangular grid ( X , Y -plane). Then a set of functions Z = F ( X , Y ) is determined, each of which represents a surface segment of the composite smooth surface. In this paper, these ideas are generalized to include a much broader class of permissible point array distributions: namely (1) the point arrangement (ordering) is topologically equivalent to a planar rectangular grid, (2) the resulting solution is a smooth composite of parametric surface segments, i.e. each surface piece is represented by a vector (point)-valued function. The solution here presented is readily applicable to a variety of problems, such as closed surface body definitions and pressure envelope surface definitions. The technique has been used successfully in these areas and others, such as numerical control milling, Newtonian impact and boundary layer.