Kinematically Redundant Octahedral Motion Platform for Virtual Reality Simulations

Nawratil, Georg; Rasoulzadeh, Arvin

doi:10.1007/978-3-319-79111-1_39

Cited by 3 publications

(2 citation statements)

References 19 publications

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“…Lebesgue lectured about Bricard's construction in 1938/39 [20], and Bennett discussed flexible octahedra in his work [6]. In recent years, there has been renewed interest in the topic: from early analysis of Bricard's octahedra [3,13,28] to their relations with a broader class of flexible surfaces [30], to applications in robotics [4,5,24], generalizations of Bricard's construction [25,26] and the analysis of flexible octahedra in different ambient spaces [23], to the study of flexibility of polyhedra via algebraic and topological techniques [1,2,10,14,22]. Flexible octahedra are self-intersecting: the first example of an embedded (i.e., with faces intersecting only at their common edges) flexible polyhedron is due to Connelly [12].…”

Section: Introductionmentioning

confidence: 99%

Combinatorics of Bricard’s octahedra

Gallet¹,

Grasegger²,

Legerský³

et al. 2021

Comptes Rendus. Mathématique

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We reprove the classification of motions of an octahedron-obtained by Bricard at the beginning of the XX century-by means of combinatorial objects satisfying some elementary rules. The explanations of these rules rely on the use of a well-known creation of modern algebraic geometry, the moduli space of stable rational curves with marked points, for the description of configurations of graphs on the sphere. Once one accepts the objects and the rules, the classification becomes elementary (though not trivial) and can be enjoyed without the need of a very deep background on the topic. Résumé. Dans cet article, on donne une preuve alternative de la classification des mouvements d'un octaèdre, originalement obtenue par Bricard au début du XX e siècle. On utilise une construction combinatoire avec un certain nombre de règles essentielles. Ces règles reposent sur une machinerie bien connue dans la géométrie algébrique moderne : l'espace de modules des courbes rationnelles stables avec des points marqués, utilisé pour codifier les configurations de graphes sur la sphère. On introduit un certain nombre d'objets et de règles : une fois que l'on les assume, la classification des mouvements d'un octaèdre telle que l'on expose devient élémentaire (bien que pas triviale) et peut être appréciée par le lecteur sans besoin de connaissances préalables très approfondies sur le sujet. 1

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Section: Introductionmentioning

confidence: 99%

Combinatorics of Bricard’s octahedra

Gallet¹,

Grasegger²,

Legerský³

et al. 2021

Comptes Rendus. Mathématique

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“…Lebesgue lectured about Bricard's construction in 1938/39 [Leb67], and Bennett discussed flexible octahedra in his work [Ben12]. In recent years, there has been renewed interest in the topic; see the works of Baker [Bak80,Bak95,Bak09], Stachel [Sta87,Sta14,Sta15], Nawratil [Naw10,NR18], and others [Con78, BS90, Mik02, Ale10, AC11, Nel10, Nel12, CY12]. The goal of this paper is to re-prove Bricard's result by employing modern techniques in algebraic geometry that hopefully may be applied to more general situations.…”

Section: Introductionmentioning

confidence: 99%

Combinatorics of Bricard's octahedra

Gallet,

Grasegger,

Legerský

et al. 2020

Preprint

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We re-prove the classification of flexible octahedra, obtained by Bricard at the beginning of the XX century, by means of combinatorial objects satisfying some elementary rules. The explanations of these rules rely on the use of a well-known creation of modern algebraic geometry, the moduli space of stable rational curves with marked points, for the description of configurations of graphs on the sphere. Once one accepts the objects and the rules, the classification becomes elementary (though not trivial) and can be enjoyed without the need of a very deep background on the topic.

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Linear Pentapods with a Simple Singularity Variety – Part I: Determination and Redundant Designs

Rasoulzadeh

Nawratil

2019

Advances in Mechanism and Machine Science

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Kinematically Redundant Octahedral Motion Platform for Virtual Reality Simulations

Cited by 3 publications

References 19 publications

Combinatorics of Bricard’s octahedra

Combinatorics of Bricard’s octahedra

Combinatorics of Bricard's octahedra

Linear Pentapods with a Simple Singularity Variety – Part I: Determination and Redundant Designs

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