Matrix and Curve Mesh Interpolation

Applications of mathematics and statistics often require fitting a smooth approximating function to a finite set of data points, or interpolating the data points by fitting the points exactly. In the event that interpolation of a grid of points is required, we recently proposed a method (which we call the SVD method[10]) for creating a function of two variables which interpolates the points. While there are many other procedures in the literature for interpolating and approximating 3-D data sets [8, 12, 13], the SVD method is straightforward and very easy to program. The process of interpolating a matrix we call skinning the matrix; we speak of creating a skin of the matrix. Sometimes a mesh or grid of interlocking functions (which we call a curve mesh) may be considered for interpolation. Curve mesh interpolation, in which this grid of interlocking functions is interpolated by a single function of two variables, has been carried out in various ways ([2, 3, 4, 14]). We use a variant of the SVD method, based on the fact that the intersections of this grid of functions create a matrix (a node matrix). We simultaneously fit this matrix and the mesh which connects them. As might be expected, we speak of a skin of a mesh. This method works for arbitrary curve meshes. We begin with the simple case of a square, invertible node-matrix meshes, and extend the technique to square, non-invertible meshes, and rectangular meshes. We present here two examples from the literature for comparison with another recent approach[2], as well as a famous example from a paper by Franke [5].