# Tracing isomanifolds in $R^d$ in time polymonial in d using Coxeter-Freudenthal-Kuhn triangulations

1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
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 from $R^d$ to $R^{d-n}$, 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 M' 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 1/D (and unavoidably exponential in n). Since it is known that for $1/D = \Omega (d^{2.5})$, M' is $O(D^2)$-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 M' 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.
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Conference papers
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https://hal.inria.fr/hal-03006663
Contributor : Jean-Daniel Boissonnat <>
Submitted on : Tuesday, March 16, 2021 - 9:15:51 AM
Last modification on : Thursday, March 18, 2021 - 3:30:53 AM

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SoCG2021-fullversion.pdf
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• HAL Id : hal-03006663, version 2

### Citation

Jean-Daniel Boissonnat, Siargey Kachanovich, Mathijs Wintraecken. Tracing isomanifolds in $R^d$ in time polymonial in d using Coxeter-Freudenthal-Kuhn triangulations. 37th Symposium on Computational Geometry, Jun 2021, Buffalo, United States. ⟨hal-03006663v2⟩

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