Isometric Point-Circle Configurations on Surfaces from Uniform Maps

Izquierdo, Milagros; Stokes, Klara

doi:10.1007/978-3-319-30451-9_10

Cited by 2 publications

(1 citation statement)

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“…If the linear realization is injective, then the incidence geometry must have the property that every pair of elements of one type is simultaneously incident with at most one element of the other type. This property has also been called linearity of the incidence geometry [10], since it captures the abstract notion of the incidences of a line arrangement.…”

Section: Graphs Configurations and Incidence Geometriesmentioning

confidence: 99%

Exploring the infinitesimal rigidity of planar configurations of points and rods

Lundqvist¹,

Stokes²,

Öhman³

2021

Preprint

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This article is concerned with the rigidity properties of geometric realizations of incidence geometries of rank two as points and lines in the Euclidean plane; we care about the distance being preserved among collinear points. We discuss the rigidity properties of geometric realizations of incidence geometries in relation to the rigidity of geometric realizations of other well-known structures, such as graphs and hypergraphs.The 2-plane matroid is also discussed. Further, we extend a result of Whiteley to determine necessary conditions for an incidence geometry of points and lines with exactly three points on each line, or 3-uniform hypergraphs, to have a minimally rigid realization as points and lines in the plane. We also give examples to show that these conditions are not sufficient.Finally, we examine the rigidity properties of v k -configurations. We provide several examples of rigid v 3 -configurations, and families of flexible geometric v 3 -configurations. The exposition of the material is supported by many figures.

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Section: Graphs Configurations and Incidence Geometriesmentioning

confidence: 99%