Title: Point-hyperplane frameworks, slider joints, and rigidity preserving transformations,
Abstract: 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 Jordán,
which deals with the case when three points are collinear.