Compact surfaces as configuration spaces of mechanical linkages

Jordan, Denis; Steiner, M

doi:10.1007/bf02809898

Cited by 8 publications

(8 citation statements)

References 7 publications

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“…II,§1.4]). Somewhat more surprisingly, every real algebraic variety is a union of components of the configuration space of some planar linkage (d = 2)see [KM3,Ki,JS]. Thus our results here appear to be statements about any real algebraic variety.…”

Section: Introductionsupporting

confidence: 50%

Generic singular configurations of linkages

Blanc

Shvalb

2012

Topology and its Applications

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We study the topological and differentiable singularities of the configuration space C(Γ) of a mechanical linkage Γ in R d , defining an inductive sufficient condition to determine when a configuration is singular. We show that this condition holds for generic singularities, provide a mechanical interpretation, and give an example of a type of mechanism for which this criterion identifies all singularities.

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Section: Introductionsupporting

confidence: 50%

Generic singular configurations of linkages

Blanc

Shvalb

2012

Topology and its Applications

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“…In particular, Theorem 1.10 combined with the work of Jordan and Steiner [JS01] yields directly Corollary 1.11. In any Riemannian surface M, the differentiable universality theorem is true for compact orientable surfaces.…”

Section: Resultsmentioning

confidence: 89%

Universality theorems for linkages in homogeneous surfaces

Kourganoff

2016

Geom Dedicata

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Abstract. A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of linkages in the (Lorentz-)Minkowski plane, in the hyperbolic plane and in the sphere. We also give a proof of a differential universality theorem in the Minkowski plane and in the hyperbolic plane: for any compact manifold M , there is a linkage whose configuration space is diffeomorphic to the disjoint union of a finite number of copies of M . In the Minkowski plane, it is also true for any manifold M which is the interior of a compact manifold with boundary. CHAPTER 1 Introduction and generalitiesA mechanical linkage is a mechanism made of rigid rods linked together by flexible joints. Mathematically, we consider a linkage as a marked graph: lengths are assigned to the edges, and some vertices are pinned down while others may move.A realization of a linkage L in a manifold M is a mapping which sends each vertex of the graph to a point of M, respecting the lengths of the edges. The configuration space Conf M (L) is the set of all realizations. Intuitively, the configuration space is the set of all the possible states of the mechanical linkage. This supposes, classically, the ambient manifold M to have a Riemannian structure: thus the configuration space may be seen as the space of "isometric immersions" of the metric graph L in M.Here we will always deal with (non-trivially) marked connected graphs, that is, a non-empty set of vertices have fixed realizations (in fact, when M is homogeneous, considering a linkage without fixed vertices only adds a translation factor to the configuration space). Hence, our configurations spaces will be compact even if M is not compact, but rather complete. Some historical backgroundMost existing studies deal with the special case where M is the Euclidean plane and some with the higher dimensional Euclidean case (see for instance [Far08] Universality theorems. When M is the Euclidean plane E 2 , a configuration space is an algebraic set. This set is smooth for a generic length structure on the underlying graph.Universality theorems tend to state that, playing with mechanisms, we get any algebraic set of R n , and any manifold, as a configuration space! In contrast, it is a hard task to understand the topology or geometry of the configuration space of a given mechanism, even for a simple one.Universality theorems have been announced in the ambient manifold E 2 by Thurston in oral lectures, and then proved by Kapovich and Millson in [KM02]. They have been proved in E n by King [Kin98], and in RP 2 and in the 2-sphere by Mnëv (see [Mnë88] and [KM02]). It is our aim in the present article to prove them in the cases of: the hyperbolic plane H 2 , the sphere S 2 and the (Lorentz-)Minkowski plane M. These are simply connected homogeneous pseudo-Riemannian surfa...

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“…In the following, we will only consider linkages such that Conf(L) is a smooth manifold in (R 2 ) n . It is the case for a generic choice of the edge lengths (see [JS01] for example).…”

Section: Mechanical Linkagesmentioning

confidence: 99%

Anosov Geodesic Flows, Billiards and Linkages

Kourganoff

2016

Commun. Math. Phys.

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Any smooth surface in R 3 may be flattened along the z-axis, and the flattened surface becomes close to a billiard table in R 2 . We show that, under some hypotheses, the geodesic flow of this surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersive and has finite horizon, then the geodesic flow of the corresponding surface is Anosov. We apply this result to the theory of mechanical linkages and their dynamics: we provide a new example of a simple linkage whose physical behavior is Anosov. For the first time, the edge lengths of the mechanism are given explicitly.

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Compact surfaces as configuration spaces of mechanical linkages

Abstract: There exists a homeomorphism between any compact orientable closed surface and the configuration space of an appropriate mechanical linkage defined by a weighted graph embedded in the Euclidean plane.

Cited by 8 publications

References 7 publications

Generic singular configurations of linkages

Generic singular configurations of linkages

Universality theorems for linkages in homogeneous surfaces

Anosov Geodesic Flows, Billiards and Linkages

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