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Tracing Isomanifolds in ℝ^d in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations

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Acknowledgements

We thank Dominique Attali, Guilherme de Fonseca, Arijit Ghosh, Vincent Pilaud and Aurélien Alvarez for their comments and suggestions. We also acknowledge the reviewers.

Cite As

Jean-Daniel Boissonnat, Siargey Kachanovich, and Mathijs Wintraecken. Tracing Isomanifolds in ℝ^d in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.17

Abstract

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art.

Subject Classification

ACM Subject Classification
• Theory of computation → Computational geometry
Keywords
• Coxeter triangulation
• Kuhn triangulation
• permutahedron
• PL-approximations
• isomanifolds/solution manifolds/isosurfacing

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