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Slider-pinning Rigidity: a Maxwell-Laman-type Theorem

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 Added by Louis Theran
 Publication date 2010
  fields
and research's language is English




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We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteleys direction networks.



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A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in $mathbb{R}^d$ and those in $mathbb{S}^d$ is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to include the infinitesimal rigidity of frameworks consisting of points and hyperplanes. This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis. Among other results, we derive a combinatorial characterization of graphs that can be realized as infinitesimally rigid frameworks in the plane with a given set of points collinear. This extends a result by Jackson and Jord{a}n, which deals with the case when three points are collinear.
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