Geometry of Point-Hyperplane and Spherical Frameworks
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In this thesis we show that the infinitesimal rigidity of point-hyperplane frameworks in Euclidean spaces is equivalent to the infinitesimal rigidity of bar-joint frameworks in spherical spaces with a set of joints (corresponding to the hyperplanes) located on a hyperplane. This is done by comparing the rigidity matrix of Euclidean point-hyperplane frameworks and the rigidity matrix of spherical frameworks. This result clearly shows how the first-order rigidity in projective spaces and Euclidean spaces are globally connected. This geometrically significant result is central to the thesis. This result leads to the equivalence of the first-order rigidity of point-hyperplane frameworks with that of bar-joint frameworks with a set of joints in a hyperplane in a Euclidean space (joint work). We also study the rigidity of point-hyperplane frameworks and characterize their rigidity in Euclidean spaces. We next highlight the relationship between point-line frameworks and slider mechanisms in the plane. Point-line frameworks are used to model various types of slider mechanisms. A combinatorial characterization of the rigidity of pinned-slider frameworks in the plane is derived directly as an immediate consequence of the analogous result for pinned bar-joint frameworks in the plane. Using fixed-normal point-line frameworks, we model a second type of slider system in which the slider directions do not change. Also, a third type of slider mechanism is introduced in which the sliders may only rotate around a fixed point but do not translate. This slider mechanism is defined using point-line frameworks with rotatory lines (no translational motion of the lines is allowed). A combinatorial characterization of the generic rigidity of these frameworks is coauthored in a joint work. Then we introduce point-hyperplane tensegrity frameworks in Euclidean spaces. We investigate the rigidity and the infinitesimal rigidity of these frameworks using tensegrity frameworks in spherical spaces. We characterize these different types of rigidity for point-hyperplane tensegrity frameworks and show how these types of rigidity are linked together. This leads to a characterization of the rigidity of a broader class of slider mechanisms in which sliders may move under variable distance constraints rather than fixed-distance constraints. Finally we investigate body-cad constraints in the plane. A combinatorial characterization of their generic infinitesimal rigidity is given. We show how angular constraints are related to non-angular constraints. This leads to a combinatorial result about the rigidity of a specific class of body-bar frameworks with point-point coincidence constraints in the space.