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.
Highlights
Given a collection of objects in a space satisfying particular geometric constraints, a fundamental question is whether the given constraints uniquely determine the whole configuration up to congruence
We establish a one-to-one correspondence between the space of infinitesimal motions of a point-hyperplane framework and that of a bar-joint framework with a given set of joints in the same hyperplane by extending the correspondence between Euclidean rigidity and spherical rigidity. Combining this with a result by Jackson and Owen [11] for point-line rigidity, we give a combinatorial characterization of a graph that can be realized as an infinitesimally rigid bar-joint framework in the plane with a given set of points collinear
We show how to derive their combinatorial characterization in the plane from the result of Jackson and Owen [11]
Summary
Given a collection of objects in a space satisfying particular geometric constraints, a fundamental question is whether the given constraints uniquely determine the whole configuration up to congruence. Combining this with a result by Jackson and Owen [11] for point-line rigidity, we give a combinatorial characterization of a graph that can be realized as an infinitesimally rigid bar-joint framework in the plane with a given set of points collinear. When EP P = ELL = ∅, we further point out a connection to the parallel drawing problem from scene analysis, and we derive a combinatorial characterization of graphs G = (VP ∪ VL, EP L) which can be realized as a fixed-normal rigid pointhyperplane framework in Rd using a theorem of Whiteley [29].
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