Natural neighbor coordinates of points on a surface

Natural neighbor coordinates and natural neighbor interpolation have been i ntroduced by Sibson for interpolating multivariate scattered data. In this paper, we consider the case where the data points belong to a smooth surfac e S, i.e., a (d - 1)-manifold of R-d. We show that the natural neighbor coo rdinates of a point X belonging to S tends to behave as a local system of c oordinates 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 dat a points converges towards the medial axis of S when the sampling density i ncreases. (C) 2001 Elsevier Science B.V All rights reserved.