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Article

Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra

1
IUMA Information and Communications System, University of Las Palmas de Gran Canaria, 35017 Canary Islands, Spain
2
Imaging Technology Center (CTIM), University of Las Palmas de Gran Canaria, 35017 Canary Islands, Spain
3
Facultad de Informática y Matemática, Universidad de Holguín, Holguín 80100, Cuba
*
Author to whom correspondence should be addressed.
Academic Editor: Javier Alcaraz
Mathematics 2021, 9(12), 1447; https://doi.org/10.3390/math9121447
Received: 22 April 2021 / Revised: 11 June 2021 / Accepted: 14 June 2021 / Published: 21 June 2021
(This article belongs to the Section Mathematics and Computer Science)
Showing whether the longest-edge (LE) bisection of tetrahedra meshes degenerates the stability condition or not is still an open problem. Some reasons, in part, are due to the cost for achieving the computation of similarity classes of millions of tetrahedra. We prove the existence of tetrahedra where the LE bisection introduces, at most, 37 similarity classes. This family of new tetrahedra was roughly pointed out by Adler in 1983. However, as far as we know, there has been no evidence confirming its existence. We also introduce a new data structure and algorithm for computing the number of similarity tetrahedral classes based on integer arithmetic, storing only the square of edges. The algorithm lets us perform compact and efficient high-level similarity class computations with a cost that is only dependent on the number of similarity classes. View Full-Text
Keywords: longest edge; bisection; similarity classes; meshes; algorithm; tetrahedra longest edge; bisection; similarity classes; meshes; algorithm; tetrahedra
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MDPI and ACS Style

Suárez, J.P.; Trujillo, A.; Moreno, T. Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra. Mathematics 2021, 9, 1447. https://doi.org/10.3390/math9121447

AMA Style

Suárez JP, Trujillo A, Moreno T. Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra. Mathematics. 2021; 9(12):1447. https://doi.org/10.3390/math9121447

Chicago/Turabian Style

Suárez, Jose P., Agustín Trujillo, and Tania Moreno. 2021. "Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra" Mathematics 9, no. 12: 1447. https://doi.org/10.3390/math9121447

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