Exact and parallel intersection of 3d triangular meshes

Abstract

This thesis presents an exact parallel algorithm for computing the intersection be- tween two 3D triangular meshes, as used in CAD/CAM (Computer Aided De- sign/Computer Aided Manufacturing), CFD (Computational Fluid Dynamics), GIS (Geographical Information Science) and additive manufacturing (also known as 3D Printing). Geometric software packages occasionally fail to compute the correct result because of the algorithm implementation complexity (that usually needs to handle several special cases) and of precision problems caused by floating point arithmetic. A failure in an intersection computation algorithm may propagate to any software using the algorithm as a subroutine. As datasets get bigger (and the chances of failure in an inexact algorithm increase), exact algorithms become even more important. While other methods for exactly intersecting meshes exist, their performance makes them non-suitable for applications where the fast processing of big geometric models is important (such as interactive CAD systems). The key to obtain robustness and performance is a combination of 5 separate techniques: • Multiple precision rational numbers, to exactly represent the coordinates of the objects and completely eliminate roundoff errors during the computations. • Simulation of Simplicity, a symbolic perturbation technique, to ensure that all geometric degeneracies (special cases) are properly handled. • Simple data representations and local information, to simplify the correct pro- cessing of the data and make the algorithm more parallelizable. • A uniform grid, to efficiently index the data, and accelerate some of the steps of the algorithm such as testing pairs of triangles for intersection or locating points in the mesh.