Abstract
The Voronoi diagram is one of the most fundamental data structure in Computational Geometry. It has also been used in Computer Aided Geometric Design for surface modelling to solve two of the most useful surface construction problems: Scattered Data Interpolation and Transfinite Surface Interpolation. Because of their complicated evaluation process, there are no formulas to compute the first and second order derivatives of those surface interpolants defined over Voronoi diagrams, which make them practically useless in any CAD/CAM (Computer Aided Design and Manufacturing) environment. In this research, we will derive those formulas for the interpolants, and give some new insights on the differentiability and curvature. We will also investigate the connection between the transfinite interpolant over some special domain and the minimal surface.
