On the Infinitesimal Rigidity of Polyhedra with Vertices in Convex Position
نویسنده
چکیده
Let P ⊂ R3 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 Hilbert-Einstein 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.
منابع مشابه
On the infinitesimal rigidity of weakly convex polyhedra
The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by “denting” at most two edges at a common vertex, and suspensions with a natu...
متن کاملSmall deformations of polygons and polyhedra
We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a quadratic invariant on the space of first-order deformations of a polygon. For convex polygons, this quadratic invariant has a positivity property, leading to a new proof of t...
متن کاملInfinitesimally Rigid Polyhedra. I. Statics of Frameworks
From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (a...
متن کاملA Rigidity Criterion for Non-Convex Polyhedra
Let P be a (non necessarily convex) embedded polyhedron in R, with its vertices on an ellipsoid. Suppose that the interior of P can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of polytopes C1, · · · , Cn with disjoint interiors, whose vertices are the vertices of P . Su...
متن کاملBending Fields, Rigidity, and the Bellows Conjecture
Aleksandrov and Pogorelov used bending fields (the velocity fields of continuous isometric deformations) to show infinitesimal rigidity of some classes of surfaces ([2] and [12]). Bending fields (and the rigidity matrix) were also successfully employed in the study of infinitesimal rigidity of polyhedra and tensegrity frameworks (see [7]). We will use these tools to re-derive a previously known...
متن کامل