Natural neighbor coordinates of points on a surface

Natural neighbor coordinates and natural neighbor interpolation have been introduced by Sibson for interpolating multivariate scattered data. In this paper, we consider the case where the data points belong to a smooth surface S, i.e., a (d - 1)-manifold of R-d. We show that the natural neighbor coordinates of a point X belonging to S tends to behave as a local system of coordinates on the surface when the density of points increases. Our result does not assume any knowledge about the ordering, connectivity or topology of the data points or of the surface. An important ingredient in our proof is the fact that a subset of the vertices of the Voronoi diagram of the data points converges towards the medial axis of S when the sampling density increases. (C) 2001 Elsevier Science B.V All rights reserved.