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Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs [chapter]

Ioannis Z. Emiris, Elias P. Tsigaridas, Antonios Varvitsiotis
2012 Distance Geometry  
A graph G is called generically minimally rigid in R d if, for any choice of sufficiently generic edge lengths, it can be embedded in R d in a finite number of distinct ways, modulo rigid transformations  ...  We explore different polynomial formulations so as to reduce the corresponding mixed volume, namely by introducing new variables that remove certain spurious roots, and by applying the theory of distance  ...  and the National Science Foundation of China (under grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation, within which part of this work was performed, and from the  ... 
doi:10.1007/978-1-4614-5128-0_2 fatcat:yprrnm2eqbbarduukkqyhj4rbq

The m-Bézout Bound and Distance Geometry [chapter]

Evangelos Bartzos, Ioannis Z. Emiris, Charalambos Tzamos
2021 Lecture Notes in Computer Science  
Our work is inspired by the application of the m-Bézout bound to counting Euclidean embeddings of distance graphs.  ...  Distance geometry and rigidity theory study graphs with a finite number of configurations, up to rigid transformations, which are prescribed by the edge lengths.  ...  The m-Bézout Bound and Distance Geometry 1 Introduction Distance Geometry is the branch of mathematics studying configurations of sets of points, when only (some of) their distances are known.  ... 
doi:10.1007/978-3-030-85165-1_2 fatcat:usaavbo3qzbcddmuwistt434ie

On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2

Evangelos Bartzos, Ioannis Z. Emiris, Jan Legerský, Elias Tsigaridas
2019 Journal of symbolic computation  
Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space R d or on a sphere and other manifolds which in addition satisfy certain edge length constraints.  ...  this classification for all 6-vertex graphs.  ...  Mixed volume techniques [15] and a recent combinatorial algorithm for Laman graphs [9] have treated the complex case.  ... 
doi:10.1016/j.jsc.2019.10.015 fatcat:uwzuztg3dja2fpapyvqnnqfruq

On the maximal number of real embeddings of minimally rigid graphs in R^2, R^3 and S^2 [article]

Evangelos Bartzos, Ioannis Z. Emiris, Jan Legerský, Elias Tsigaridas
2018 arXiv   pre-print
Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space R^d or on a sphere and which in addition satisfy certain edge length constraints.  ...  this classification for all 6-vertex graphs.  ...  Acknowledgments This work is part of the project ARCADES that has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement  ... 
arXiv:1811.12800v1 fatcat:ak3ahn6qszhvxczshtxkvgji3i

On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs [article]

Evangelos Bartzos and Ioannis Z. Emiris and Josef Schicho
2020 arXiv   pre-print
Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d≥ 5, both in the Euclidean and spherical case.  ...  Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S^2 and C^3.  ...  We thank Georg Grasegger for providing us with runtimes of the combinatorial algorithm that calculates c d (G) and for explaining to EB the counting of embeddings for triangle-free Geiringer graphs.  ... 
arXiv:2005.14485v2 fatcat:hfrwv4en3zcatorbb3pcgb35yy

Euclidean distance geometry and applications [article]

Leo Liberti, Carlile Lavor, Nelson Maculan, Antonio Mucherino
2012 arXiv   pre-print
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance.  ...  We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.  ...  We also wish to thank Chiara Bellasio for providing inspiring dishes, a pleasant atmosphere and lots of patience and support during many working  ... 
arXiv:1205.0349v1 fatcat:jpr2dxrcj5a6degsanug3bbhcu

Euclidean Distance Geometry and Applications

Leo Liberti, Carlile Lavor, Nelson Maculan, Antonio Mucherino
2014 SIAM Review  
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance.  ...  We survey the theory of Euclidean distance geometry and its most important applications, with special emphasis on molecular conformation problems.  ...  The graph embedding problem on a given manifold, in the topological sense, is the problem of finding a topological embedding for a graph in the manifold: the constraints are given not by the distances,  ... 
doi:10.1137/120875909 fatcat:imd7wulmn5hgjgx3ty6db7j6bm

Mixed Volume Techniques for Embeddings of Laman Graphs [article]

Reinhard Steffens, Thorsten Theobald
2009 arXiv   pre-print
While in most cases the resulting bounds are weaker than the best known bounds on the number of embeddings, for some classes of graphs the bounds are tight.  ...  The focus of the paper is to provide the methods to study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints.  ...  Given a Laman graph framework we want to know how many embeddings, i.e. maps α : V → R 2 , exist such that the Euclidean distance between two points in the image is exactly l i,j for all [v i , v j ] ∈  ... 
arXiv:0805.4120v2 fatcat:7c33nmzhmvgndkyu66gbqd7kmm

Open Research Areas in Distance Geometry [chapter]

Leo Liberti, Carlile Lavor
2018 Variational Analysis and Aerospace Engineering  
Distance Geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances.  ...  We briefly introduce the field, and discuss some open and promising research areas.  ...  Acknowledgments We are grateful for Ana Flavia Lima for helping to check the manuscript prior to submission. LL was partly supported by the ANR "Bip:Bip" project n. ANR-10-BINF-03-08.  ... 
doi:10.1007/978-3-319-99142-9_11 fatcat:zw2pcbyedzg6nflwrpmw6ytcgy

Open research areas in distance geometry [article]

Leo Liberti, Carlile Lavor
2016 arXiv   pre-print
Distance Geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances.  ...  We briefly introduce the field, and discuss some open and promising research areas.  ...  Acknowledgments We are grateful for Ana Flavia Lima for helping to check the manuscript prior to submission. LL was partly supported by the ANR "Bip:Bip" project n. ANR-10-BINF-03-08.  ... 
arXiv:1610.00652v1 fatcat:qxmc3okucvczrb3ki5j2oq6cdu

Mixed volume techniques for embeddings of Laman graphs

Reinhard Steffens, Thorsten Theobald
2010 Computational geometry  
While in most cases the resulting bounds are weaker than the best known bounds on the number of embeddings, for some classes of graphs the bounds are tight.  ...  The focus of the paper is to provide methods to study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints.  ...  Determining the maximal number of embeddings (modulo rigid motions) for a given Laman graph is an open problem.  ... 
doi:10.1016/j.comgeo.2009.04.004 fatcat:jb2gawumgvcwbexilycok2gydy

Symmetry in 3D Geometry: Extraction and Applications

Niloy J. Mitra, Mark Pauly, Michael Wand, Duygu Ceylan
2013 Computer graphics forum (Print)  
Numerous methods have been proposed to find, extract, encode, and exploit geometric symmetries and high-level structural information for a wide variety of geometry processing tasks.  ...  We discuss a variety of applications in computer graphics and geometry processing that benefit from symmetry information for more effective processing.  ...  The authors thank Hao (Richard) Zhang and the anonymous reviewers for their comments on various versions of this manuscript.  ... 
doi:10.1111/cgf.12010 fatcat:mbgf3fe7qjajlkkoiiucqfkoaa

Computationally Tractable Riemannian Manifolds for Graph Embeddings [article]

Calin Cruceru, Gary Bécigneul, Octavian-Eugen Ganea
2020 arXiv   pre-print
Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.  ...  Empirically, we demonstrate consistent improvements over Euclidean geometry while often outperforming hyperbolic and elliptical embeddings based on various metrics that capture different graph properties  ...  Acknowledgements We would like to thank Andreas Bloch for suggesting the curvature quantification approach based on the sum of angles in geodesic triangles. We are grateful to Prof.  ... 
arXiv:2002.08665v2 fatcat:oxafwmwkrvd33nf3c72mr4yn3y

New upper bounds for the number of embeddings of minimally rigid graphs [article]

Evangelos Bartzos and Ioannis Z. Emiris and Raimundas Vidunas
2021 arXiv   pre-print
By definition, a rigid graph in ℝ^d (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints.  ...  For the most important cases of d=2 and d=3, the new bounds are O(3.7764^|V|) and O(6.8399^|V|), respectively.  ...  EB and IZE are members of team AROMATH, joint between INRIA Sophia-Antipolis, France, and NKUA.  ... 
arXiv:2010.10578v2 fatcat:tebye4r44rekpkjwu3v5eebd5y

Semi-bipartite Graph Visualization for Gene Ontology Networks [chapter]

Kai Xu, Rohan Williams, Seok-Hee Hong, Qing Liu, Ji Zhang
2010 Lecture Notes in Computer Science  
Algebraic methods for counting Euclidean embeddings of rigid graphs Ioannis Emiris, Elias Tsigaridas, and Antonios Varvitsiotis Abstract: The study of (minimally) rigid graphs is motivated by numerous  ...  A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space.  ... 
doi:10.1007/978-3-642-11805-0_24 fatcat:724qxxufdzgthfzrovmnj4nzky
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