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Point-hyperplane frameworks, slider joints, and rigidity preserving transformations

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<mark>Journal publication date</mark>03/2019
<mark>Journal</mark>Journal of Combinatorial Theory, Series B
Volume135
Number of pages31
Pages (from-to)44-74
Publication statusPublished
Early online date2/08/18
Original languageEnglish

Abstract

A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in Rd and those in Sd 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 Jordán, which deals with the case when three points are collinear.