What is it about?
Tetrahedra are four-sided pyramids with triangular bases. They can be used as building blocks for any three-dimensional shape. The intersection between two tetrahedra is important in a number of applications, but is not always straightforward to compute numerically. This article presents a robust algorithm for this computation, meaning any small numerical error in the calculations will not significantly affect the outcome. The algorithm uses only those calculations that are necessary to produce a result, making use of as much prior information as possible.
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Why is it important?
3D intersections can be important in computer graphics, when two objects collide. They also appear in some numerical methods involving finite elements, where large domains are split up into triangulations. When multiple triangulations overlap, information from one needs to pass to the other, which requires intersecting several tetrahedra.
Perspectives
This builds off of our previous work in intersecting triangles. Further generalization will lead to the intersection of simplices, which are higher dimensional versions of tetrahedra. Many of the ideas in the article, such as parsimony and binary-valued sign functions, may offer new possibilities to other geometric algorithms.
Conor McCoid
Universite de Geneve
Read the Original
This page is a summary of: Intersection of tetrahedra, ACM Transactions on Mathematical Software, April 2025, ACM (Association for Computing Machinery),
DOI: 10.1145/3729532.
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