In 2D, there are two generic triangles, namely the acute and obtuse triangles, the transitional case being the right angle triangle. This classification can be phrased in terms of critical points of the Euclidean distance function to the vertices of the triangle. This point of view offers a strategy for investigating classifications and transitions in more general settings, and this paper is concerned with the 3D Euclidean and power distances cases.
In the Euclidean case, elaborating upon the classification of tetrahedra recently published, we establish the set of feasible transitions. Regarding the power distance, our results are twofold. First, we provide conjectures on the the classification of tetrahedra and the associated transitions. Second, we provide the algebraic specification of the classification problem, using an encoding of the tetrahedra based on edges’ lengths. This latter contribution calls for future work in computational algebraic geometry so as to certify our conjectures, and might also allow one to extend our approach in higher dimension.
We expect these insights to find applications wherever the shape of simplices matters, for example in meshing and geometry processing.
Joint work with Nisarg Shah and Sushant Sachdeva.
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