Exact and Efficient Polyhedral Envelope Containment Check

We introduce a new technique to check containment of a triangle within an envelope built around a given triangle mesh. While existing methods conservatively check containment within a Euclidean envelope, our approach makes use of a non-Euclidean envelope where containment can be checked both exactly and efficiently. Exactness is crucial to address major robustness issues in existing geometry processing algorithms, which we demonstrate by integrating our technique in two surface triangle remeshing algorithms and a volumetric tetrahedral meshing algorithm. We provide a quantitative comparison of our method and alternative algorithms, showing that our solution, in addition to being exact, is also more efficient. Indeed, while containment within large envelopes can be checked in a comparable time, we show that our algorithm outperforms alternative methods when the envelope becomes thin.

For paper and source code, please visit: https://github.com/wangbolun300/fast-envelope.

This work was supported in part through the NYU IT High Performance Computing resources, services, and staff expertise. This work was partially supported by the NSF CAREER award under Grant No. 1652515, the NSF grants OAC-1835712, OIA-1937043, CHS-1908767, CHS-1901091, National Key Research and Development Program of China No. 2018YFB1107402, EU ERC Advanced Grant CHANGE No. 694515, a gift from Adobe Research, a gift from nTopology, and a gift from Advanced Micro Devices, Inc.