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27
Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds
, 2009
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Principal curvatures from the integral invariant viewpoint
, 2007
"... The extraction of curvature information for surfaces is a basic problem of Geometry Processing. Recently an integral invariant solution of this problem was presented, which is based on principal component analysis of local neighbourhoods defined by kernel balls of various sizes. It is not only robus ..."
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Cited by 12 (3 self)
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The extraction of curvature information for surfaces is a basic problem of Geometry Processing. Recently an integral invariant solution of this problem was presented, which is based on principal component analysis of local neighbourhoods defined by kernel balls of various sizes. It is not only robust to noise, but also adjusts to the level of detail required. In the present paper we show an asymptotic analysis of the moments of inertia and the principal directions which are used in this approach. We also address implementation and, briefly, robustness issues and applications.
A VARIATIONAL PRINCIPLE FOR WEIGHTED DELAUNAY TRIANGULATIONS AND HYPERIDEAL Polyhedra
, 2008
"... We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with nonintersecting sitecircles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperb ..."
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Cited by 11 (0 self)
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We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with nonintersecting sitecircles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus, the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, we consider weighted Delaunay triangulations in piecewise flat surfaces, allowing cone singularities with prescribed cone angles in the vertices. The material presented here extends work by Rivin on Delaunay triangulations and ideal polyhedra.
From Spaces of Polygons to Spaces of Polyhedra Following Bavard, Ghys and Thurston
, 2009
"... After work of W. P. Thurston, C. Bavard and É. Ghys constructed particular hyperbolic polyhedra from spaces of deformations of Euclidean polygons. We present this construction as a straightforward consequence of the theory of mixedvolumes. The gluing of these polyhedra can be isometrically embedded ..."
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Cited by 8 (1 self)
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After work of W. P. Thurston, C. Bavard and É. Ghys constructed particular hyperbolic polyhedra from spaces of deformations of Euclidean polygons. We present this construction as a straightforward consequence of the theory of mixedvolumes. The gluing of these polyhedra can be isometrically embedded into complex hyperbolic conemanifolds constructed by Thurston from spaces of deformations of Euclidean polyhedra. It is then possible to deduce the metric structure of the spaces of polygons embedded in complex hyperbolic orbifolds discovered by P. Deligne and G. D. Mostow.
ON THE INFINITESIMAL RIGIDITY OF POLYHEDRA WITH VERTICES IN CONVEX POSITION
"... Abstract. Let P ⊂ R 3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a wea ..."
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Cited by 7 (2 self)
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Abstract. Let P ⊂ R 3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additionnal assumption of codecomposability. The proof relies on a result of independent interest concerning the HilbertEinstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite. 1.
The Colin de Verdière number and graphs of polytopes
, 2008
"... The Colin de Verdière number µ(G) of a graph G is the maximum corank of a Colin de Verdière matrix for G (that is, of a Schrödinger operator on G with a single negative eigenvalue). In 2001, Lovász gave a construction that associated to every convex 3polytope a Colin de Verdière matrix of corank 3 ..."
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Cited by 6 (2 self)
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The Colin de Verdière number µ(G) of a graph G is the maximum corank of a Colin de Verdière matrix for G (that is, of a Schrödinger operator on G with a single negative eigenvalue). In 2001, Lovász gave a construction that associated to every convex 3polytope a Colin de Verdière matrix of corank 3 for its 1skeleton. We generalize the Lovász construction to higher dimensions by interpreting it as minus the Hessian matrix of the volume of the polar dual. As a corollary, µ(G) ≥ d if G is the 1skeleton of a convex dpolytope. Determination of the signature of the Hessian of the volume is based on the second Minkowski inequality for mixed volumes and on Bol’s condition for equality.
A variational proof of Alexandrov’s convex cap theorem; arXiv: math/0703169. [333
"... We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are ..."
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Cited by 6 (3 self)
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We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that is uniquely determined by their curvatures. 1
Hyperbolic cusps with convex polyhedral boundary
 Geom. Topol
"... We prove that a 3–dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex ..."
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Cited by 6 (5 self)
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We prove that a 3–dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp. The proof uses the discrete total curvature functional on the space of “cusps with particles”, which are hyperbolic conemanifolds with the singular locus a union of halflines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus. Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces. 57M50; 53C24 1
Polyhedral hyperbolic metrics on surfaces
 Geom. Dedicata
"... Abstract. Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space 3 and a group G of ..."
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Cited by 5 (3 self)
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Abstract. Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space 3 and a group G of isometries of 3 such that the induced metric on the quotient P/G is isometric to g. Moreover, the pair (P, G) is unique among a particular class of convex polyhedra.