The Deformation Space of Geodesic Triangulations and Generalized Tutte's Embedding Theorem

Luo, Yanwen; Wu, Tianqi; Zhu, Xiaoping

doi:10.48550/arxiv.2105.00612

Cited by 6 publications

(20 citation statements)

References 8 publications

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“…This paper is a continuation of the previous work [13], where we proved that the deformation space of geodesic triangulations of a surface with negative curvature is contractible. The purpose of this paper is to identify the homotopy type of the deformation space of geodesic triangulations of a flat torus.…”

Section: Introductionmentioning

confidence: 51%

“…This paper affirms this conjecture in the case of flat tori. The case of hyperbolic surfaces has been proved in [13]. In a very recent work, Erickson-Lin [8] proved independently a generalized version of our Theorem 1.1 for general graph drawings on a flat torus.…”

Section: Introductionmentioning

confidence: 94%

“…Similar to the previous work [13], our key idea to prove Theorem 1.1 originates from Tutte's embedding theorem.…”

Section: Introductionmentioning

confidence: 98%

“…The study of the homotopy types of spaces of geodesic triangulations stemmed from the work by Cairns [2]. A brief history of this problem can be found in [13]. These spaces are closely related to diffeomoprhism groups of surfaces.…”

Section: Introductionmentioning

confidence: 99%

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The Deformation Spaces of Geodesic Triangulations of Flat Tori

Luo,

Wu,

Zhu

2021

Preprint

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We prove that the deformation space of geodesic triangulations of a flat torus is homotopy equivalent to a torus. This solves an open problem proposed by Connelly et al. in 1983, in the case of flat tori. A key tool of the proof is a generalization of Tutte's embedding theorem for flat tori. When this paper is under preparation, Erickson and Lin proved a similar result, which works for all convex drawings.

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Section: Introductionmentioning

confidence: 51%

Section: Introductionmentioning

confidence: 94%

“…Similar to the previous work [13], our key idea to prove Theorem 1.1 originates from Tutte's embedding theorem.…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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The Deformation Spaces of Geodesic Triangulations of Flat Tori

Luo,

Wu,

Zhu

2021

Preprint

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“…This conjecture has been confirmed by Bloch-Connelly-Henderson [2] for convex polygons, and a new proof based on Tuttes' embedding theorem was provided by Luo [13]. Recently, this conjecture was proved for the cases of flat tori and closed surfaces of negative curvature (see Erickson-Lin [10] and Luo-Wu-Zhu [14,15]).…”

Section: Figure 1 the Edge Invariantmentioning

confidence: 82%

The Deformation Space of Delaunay Triangulations of the Sphere

Luo¹,

Wu²,

Zhu³

2022

Preprint

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In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit 2-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit 2-sphere. These spaces can be regarded as discretized groups of diffeomorphisms of the unit 2-sphere. Hence, it is natural to conjecture that these spaces have the same homotopy types as those of their smooth counterparts. The main result of this paper confirms this conjecture for the unit 2-sphere. It follows from an observation on the variational principles on triangulated surfaces developed by I. Rivin.On the contrary, the similar conjecture does not hold in the cases of flat tori and convex polygons. We will construct simple examples of flat tori and convex polygons such that the corresponding spaces of Delaunay geodesic triangulations are not connected.

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Planar and Toroidal Morphs Made Easier

Erickson¹,

Lin²

2021

Preprint

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We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater's asymmetric extension of Tutte's classical spring-embedding theorem.First, we give a much simpler algorithm to construct piecewise-linear morphs between planar straight-line graphs. Specifically, given isomorphic straight-line drawings Γ 0 and Γ 1 of the same 3-connected planar graph G, with the same convex outer face, we construct a morph from Γ 0 to Γ 1 that consists of O(n) unidirectional morphing steps, in O(n 1+ω/2 ) time. Our algorithm entirely avoids the classical edge-collapsing strategy dating back to Cairns; instead, in each morphing step, we interpolate the pair of weights associated with a single edge.Second, we describe a natural extension of barycentric interpolation to geodesic graphs on the flat torus. Barycentric interpolation cannot be applied directly in this setting, because the linear systems defining intermediate vertex positions are not necessarily solvable. We describe a simple scaling strategy that circumvents this issue. Computing the appropriate scaling requires O(n ω/2 ) time, after which we can can compute the drawing at any point in the morph in O(n ω/2 ) time. Our algorithm is considerably simpler than the recent algorithm of Chambers et al. and produces more natural morphs. Our techniques also yield a simple proof of a conjecture of Connelly et al. for geodesic torus triangulations.

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The Deformation Space of Geodesic Triangulations and Generalized Tutte's Embedding Theorem

Cited by 6 publications

References 8 publications

The Deformation Spaces of Geodesic Triangulations of Flat Tori

The Deformation Spaces of Geodesic Triangulations of Flat Tori

The Deformation Space of Delaunay Triangulations of the Sphere

Planar and Toroidal Morphs Made Easier

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