Numerical Algorithms 5 (1993) 63-70 63 An algorithm for natural spline interpolation Leonardo Traversoni Departamento de Ingenieria de Procesos e Hidraulica, Division de Cieneias Basicas e Ingenieria, Universidad Autonoma Metropolitana, Iztapalapa, Mexico D.F., Mexico Based on the work of Robin Sibson concerning Natural Neighbor Interpolant, this paper is devoted to incorporate this concept in Spline theory. To do this, first a new concept, the "Covering Spheres", is presented, which is then linked with Sibson's interpolant. Finally, the interpolant is reformulated to present it as a Bernstein polynomial in local coordinates instead of the usual presentation as rational quartics. As a corollary, the whole idea is presented as modified Vertex Splines. 1. Natural neighbors Natural neighbors may be defined as follows: Let V = {Pl,P2, ...,Pro} be a set ofm different points on E. For each i on Im we define the set Ti = {xeE : d(x,pi)<d(x, pj),jeIm\{i}}, called a Voronoipolytope of the pointp/with respect to V. Every pair of points whose Voronoi polytopes have common frontiers are called naturalneighbors; for each point there always exists a set of natural neighbors. Two points Pi and pj are natural neighbors if they have at least a common face or side of their Voronoi polytopes. The interpolation in some point P is performed using its natural neighbors as follows. Let P be a point not belonging to V; however it may be added to V therefore modifying the Voronoi tessellation to include P on V. In that case, there will be a set of points of V which will be natural neighbors of P. If each point of V has an associated value of a certain variable, the value of the variable at P may be defined using the values in its natural neighbors. Let V = V1, V2, .... , Vm be the subset of V formed by the natural neighbors of P and let Im = zl, z2, ..... , zm be some assigned values of a function U(z). In the above points, 9 J.C. Baltzer AG, Science Publishers