Geometric Modeling Based on Polygonal Meshes

This document is the support of a SIGGRAPH 2007 course.In the last years triangle meshes have become increasingly popular and are nowadays intensively used in many different areas of computer graphics and geometry processing. In classical CAGD irregular triangle meshes developed into a valuable alternative to traditional spline surfaces, since their conceptual simplicity allows for more flexible and highly efficient processing. Moreover, the consequent use of triangle meshes as surface representation avoids error-prone conversions, e.g., from CAD surfaces to mesh-based input data of numerical simulations. Besides classical geometric modeling, other major areas frequently employing triangle meshes are computer games and movie production. In this context geometric models are often acquired by 3D scanning techniques and have to undergo post-processing and shape optimization techniques before being actually used in production. This course discusses the whole geometry processing pipeline based on triangle meshes. We will first introduce general concepts of surface representations and point out the advantageous properties of triangle meshes in Chapter 2, and present efficient data structures for their implementation in Chapter 3. The different sources of input data and types of geometric and topological degeneracies and inconsistencies are described in Chapter 4, as well as techniques for their removal, resulting in clean two-manifold meshes suitable for further processing. Mesh quality criteria measuring geometric smoothness and element shape together with the corresponding analysis techniques are presented in Chapter 6. Mesh smoothing reduces noise in scanned surfaces by generalizing signal processing techniques to irregular triangle meshes (Chapter 7). Similarly, the underlying concepts from differential geometry are useful for surface parametrization as well (Chapter 8). Due to the enormous complexity of meshes acquired by 3D scanning, mesh decimation techniques are required for error-controlled simplification (Chapter 9). The shape of triangles, which is important for the robustness of numerical simulations, can be optimized by general remeshing methods (Chapter 10). After optimizing meshes with respect to the different quality criteria, we finally present techniques for intuitive and interactive shape deformation (Chapter 11). Since solving linear systems is a commonly required component for many of the presented mesh processing algorithms, we will discuss their efficient solution and compare several existing libraries in Chapter 12