On the moduli space of polygons in the Euclidean plane

Kapovich, Michael; Millson, John J.

doi:10.4310/jdg/1214457237

Cited by 107 publications

(164 citation statements)

References 15 publications

(20 reference statements)

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“…[4] or [8], that the set of non-singular configuration spaces of a planar 5-polygon contains Eg with g _< 4. Starting with a simple 5-polygon for which the configuration space is a toms, we add two edges connected by a link to increase the genus of the surface by one.…”

Section: Theorem 11: Let ~G Be Any Compact Orientable Closed Surfacementioning

confidence: 99%

Compact surfaces as configuration spaces of mechanical linkages

Jordan

Steiner

2001

Isr. J. Math.

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Section: Theorem 11: Let ~G Be Any Compact Orientable Closed Surfacementioning

confidence: 99%

Compact surfaces as configuration spaces of mechanical linkages

Jordan

Steiner

2001

Isr. J. Math.

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“…Polygon spaces were studied by K Walker [13], M Kapovich and J Millson [9] and others. Betti numbers of the spaces N`were first described by A A Klyachko [11] who used methods of algebraic geometry.…”

Section: So2/: (4)mentioning

confidence: 99%

Topology of random linkages

Farber

2008

Algebr. Geom. Topol.

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numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters -the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R 3 . We also prove results about higher moments of Betti numbers.55R80, 55N99; 55M99

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“…Conventionally, loop closure constraints have been formulated as equality constraints (of highly non-linear functions) over joint parameters. This formulation shows that for generic linkages the set of closure configurations is a smooth submanifold of the ambient joint parameter space, and in many cases its topology is partly or completely known (see, e.g., [5,6] and [7,8], which treat planar and spatial linkages with spherical-type joints).…”

Section: Introductionmentioning

confidence: 99%

Simplex-Tree Based Kinematics of Foldable Objects as Multi-body Systems Involving Loops

Han

Rudolph

2008

Robotics: Science and Systems IV

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Abstract-Many practical multi-body systems involve loops. Studying the kinematics of such systems has been challenging, partly because of the requirement of maintaining loop closure constraints, which have conventionally been formulated as highly nonlinear equations in joint parameters. Recently, novel parameters defined by trees of triangles have been introduced for a broad class of linkage systems involving loops (e.g., spatial loops with spherical joints and planar loops with revolute joints); these parameters greatly simplify kinematics related computations and endow system configuration spaces with highly tractable piecewise convex geometries. In this paper, we describe a more general approach for multi-body systems, with loops, that allow construction trees of simplices. We illustrate the applicability and efficiency of our simplex-tree based approach to kinematics by a study of foldable objects. We present two sets of new parameters for single-vertex rigid fold kinematics; like the parameters in the triangle-tree prototype, each has a geometrically meaningful and computationally tractable constraint formulation, and each endows the configuration space with a nice geometry.

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On the moduli space of polygons in the Euclidean plane

Abstract: We study the topology of moduli spaces of polygons with fixed side lengths in the Euclidean plane. We establish a duality between the spaces of marked Euclidean polygons with fixed side lengths and marked convex Euclidean polygons with prescribed angles.

Cited by 107 publications

References 15 publications

Compact surfaces as configuration spaces of mechanical linkages

Compact surfaces as configuration spaces of mechanical linkages

Topology of random linkages

Simplex-Tree Based Kinematics of Foldable Objects as Multi-body Systems Involving Loops

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