Transfinite surface interpolation over Voronoi diagrams

Current methods for the construction of surfaces through boundary curves (transfinite surface interpolation) are limited to three or four boundary curves. This dissertation presents a new transfinite surface interpolation scheme which allows any number of input boundary curves to be arranged in nearly any configuration. The resulting surfaces are guaranteed to lie within the minimal box (convex hull) containing the input curves, and closely resemble minimal surfaces. In addition, this method can interpolate surfaces with holes (where each hole has boundary curves of its own), boundaries which do or do not form a closed loop, and parametrically defined boundary curves (where the surface may self-intersect) with arbitrarily-shaped surface domains. Scattered points and line segments with associated surface values can also be included in the boundary curve interpolation process.; This new method, the transfinite Sibson's interpolant, is an extension of a discrete data interpolant first proposed by R. Sibson. The discrete Sibson's interpolant is based on the ratios of certain subtile areas within Voronoi diagrams (or Dirichlet tessellations or Thiessen diagrams) of the input scattered data points in the plane. These ratios are scaled by values associated with each data point and define a surface which interpolates to the values. Similarly, the transfinite Sibson's interpolant is based on the ratios of certain subtile areas within Voronoi diagrams of the input boundary curves in the surface's domain. The final interpolated surface points are computed from integrals of these ratios multiplied by functions associated with each boundary curve in an analogous manner to the discrete case.; A great deal of future research is possible on this topic because this method, as well as the discrete Sibson's interpolant, can theoretically be extended to interpolate data of any dimension. The transfinite case can also be easily parallelized to decrease computation times.