Multi-dimensional Hermite Interpolation and Approximation for Modelling and Visualization

In this paper we use some well known theorems of algebraic geometry in reducing polynomial Hermite interpolation and approximation in any dimension to the solution of linear systems. We present a mix of symbolic and numerical algorithms for low degree curve ts through points in the plane, surface ts through points and curves in space, and in general, hypersuface ts through points, curves, surfaces, and sub-varieties in n dimensional space. These interpolatory and (or) approximatory ts may also be made to match derivative information along all the sub varieties. Such multi-dimensional hypersurface interpolation and approximation provides mathematical models for scattered data sampled in three or higher dimensions and can be used to compute volumes, gradients, or more uniform samples for easy and realistic visualization.