Representing implicit surfaces satisfying Lipschitz conditions by 4-dimensional point sets

Abstract

For any given implicit surface satisfying two Lipschitz conditions, this work triangulates the surface using a set of 4-dimensional points with small Hausdorff distances. Every 4-dimensional point is a 3-dimensional point with a weight. Compared to traditional triangulation approaches, our method does not explicitly require the storage of connectivity information. Instead, the connectivity information can be implicitly generated using a concept called the alpha shape. With two calculated bounds, the density of the 4-dimensional points adapts to the local curvature. Moreover, the generated triangulation has been mathematically proven to be homeomorphic to the original implicit surface. The Hausdorff distance between the triangulation and the original surface is demonstrated to be small enough according to the experimental results.