Chapter 2 - Fundaments for free-form surfaces

Abstract

An overview of the foundations of free-form surfaces is given, beginning with the three main steps for surface characterization: surface representation, surface decomposition, and surface analytics. Using Gauss's Theorema Egregium the main differences between Euclidean and non-Euclidean surfaces are explained. Given the requirements for free-form surfaces, the different ways to represent free-form surfaces are given for both discrete and continuous representations. Sampling strategies (extraction) of free-form surfaces and its dual reconstruction from sampled points of free-form surface are introduced. Free-form surface decomposition is discussed, which divides the free-form surface into various scales, starting with determining a reference form surface, through to either optimized fitting if the nominal form is known, or using special filters if the nominal form is not known. Decomposition of the free-form surface into elements at differing scales using filters defined on non-Euclidean surface is then described: diffusion filters using a PDE, Laplace-Beltrami operator, morphological filters, segmentation, and wavelet filters. The scales of interest can then be reconstructed to create the surfaces of interest for further analysis. Finally, free-form analytics is introduced that includes form characterization, shape characterization, and texture characterization, the latter being subdivided into surface field parameters and surface feature parameters as in areal surface texture.