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Duality structures and discrete conformal variations of piecewise constant curvature surfaces
"... Abstract. A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry w ..."
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Abstract. A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Additional structure is defined either by giving a geometric structure to the Poincare ́ dual of the triangulation or by assigning a discrete metric, a way of assigning length to oriented edges. This notion leads to a notion of discrete conformal structure, generalizing the discrete conformal structures based on circle packings and their generalizations studied by Thurston and others. We define and analyze conformal variations of piecewise constant curvature 2manifolds, giving particular attention to the variation of angles. We give formulas for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures. Our formulas allow us to identify particular curvature functionals associated with conformal variations. Finally, we provide a complete classification of discrete conformal structures in each of the background geometries. 1.
SHAPES OF POLYHEDRA, MIXED VOLUMES, AND HYPERBOLIC GEOMETRY
"... Abstract. We are generalizing to higher dimensions the BavardGhys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex ddimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to ..."
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Abstract. We are generalizing to higher dimensions the BavardGhys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex ddimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The AlexandrovFenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting conemanifold are equal to pi