Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

Bonnard, Bernard; Cots, Olivier; Jassionnesse, Lionel

doi:10.1007/978-3-319-02132-4_4

Cited by 9 publications

(7 citation statements)

References 15 publications

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“…Of particular relevance to this paper is the so-called "last geometric statement of Jacobi", which asserts (among other things) that the conjugate locus of a non-umbilic point on the triaxial ellipsoid has precisely 4 cusps (see [20] for a historical sketch and list of references). This conjecture was recently proved by Itoh and Kiyohara [13], and a renewed interest in the conjugate and cut locus can be seen in the recent papers providing formal studies ( [11], [24], [14], [15]), simulations ( [23], [8], [21], [22], [20]) and applications ( [6], [5], [3], [4], [7]). It is no surprise that the papers which focused on the triaxial ellipsoid and surfaces of revolution made heavy use of the fact that the geodesic flow on those surfaces is (Liouville) integrable.…”

Section: Introductionmentioning

confidence: 95%

Bifurcations of the conjugate locus

Waters

2017

Journal of Geometry and Physics

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The conjugate locus of a point $p$ in a surface $\mathcal{S}$ will have a certain number of cusps. As the point $p$ is moved in the surface the conjugate locus may spontaneously gain or lose cusps. In this paper we explain this `bifurcation' in terms of the vanishing of higher derivatives of the exponential map; we derive simple equations for these higher derivatives in terms of scalar invariants; we classify the bifurcations of cusps in terms of the local structure of the conjugate locus; and we describe an intuitive picture of the bifurcation as the intersection between certain contours in the tangent plane.Comment: Accepted in Journal of Geometry and Physics April 201

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Section: Introductionmentioning

confidence: 95%

Bifurcations of the conjugate locus

Waters

2017

Journal of Geometry and Physics

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“…The numerical approximation of the cut locus of p has seen only sparse and diverse attempts in the past. For manifolds for which explicit parametrizations are available, the cut locus can be found by studying where geodesics collide [43] or studying conjugate points and Jacobi fields [7,10] (we refer the reader to these two articles for the definition of such mathematical objects). However, when triangulated manifolds are considered, it is hard to extend these approaches.…”

Section: Introductionmentioning

confidence: 99%

Computing the cut locus of a Riemannian manifoldviaoptimal transport

Facca

Berti

Fassò³

et al. 2022

ESAIM: M2AN

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In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver, based on the so-called dynamical Monge-Kantorovich approach, we propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. We show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in $R^{3}$, and discuss advantages and limitations.

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“…Let S be a real analytic surface without boundary embedded in R 3 , and b ∈ S any point of S (that can be thought of as a base point).…”

Section: Introductionmentioning

confidence: 99%

“…We may also mention [3], where the authors use some more geometric tools to compute (numerically) the cut locus of an ellipsoid, or a sphere with some particular metric with singularities.…”

Section: Introductionmentioning

confidence: 99%