A Rigidity Criterion for Non-Convex Polyhedra

Let P be a (non-necessarily convex) embedded polyhedron in R³, with its vertices on the boundary of 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, . . ., C_n with disjoint interiors, whose vertices are the vertices of P. Suppose that there exists an ellipsoid which contains no vertex of P but intersects all the edges of the C_i. Then P is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra.     

Infinitesimal Rigidity of Convex Polytopes

Aleksandrov [1] proved that a simple convex d -dimensional polytope, d ≥ 3 , is infinitesimally rigid if the volumes of its facets satisfy a certain assumption of stationarity. We extend this result by proving that this assumption can be replaced by a stationarity assumption on the k -dimensional volumes of the polytope's k -dimensional faces, where k ∈{1,. . .,d-1} . Received November 20, 1997.     

Rigidity and polarity

Any projective polarity in 3-space transforms a statically (or equivalently, infinitesimally) rigid bar-and-joint framework into a statically rigid hinged sheetwork — a set of plane-statically rigid sheets, joined in pairs along hinge lines. In the more general class of jointed sheetworks, which is closed under polarity, static rigidity is also preserved by the polarities. In particular, the class of infinitesimally (or statically) rigid polyhedra, built with joints at the vertices and bars triangulating the faces, is closed under polarity.Work supported, in part, by grants from N.S.E.R.C. (Canada) and F.C.A.C. (Quebec), and a visiting appointment at McGill University.     

A spatial unit-bar-framework which is rigid and triangle-free

We present an infinitesimally rigid unit-bar-framework in 3-space which contains no triangle.     

Robust Tensegrity Polygons

A tensegrity polygon is a planar cable-strut tensegrity framework in which the cables form a convex polygon containing all vertices. The underlying edge-labeled graph $T=(V;C,S)$ T = ( V ; C , S ) , in which the cable edges form a Hamilton cycle, is an abstract tensegrity polygon. It is said to be robust if every convex realization of T as a tensegrity polygon has an equilibrium stress which is positive on the cables and negative on the struts, or equivalently, if every convex realization of T is infinitesimally rigid. We characterize the robust abstract tensegrity polygons on n vertices with $n-2$ n - 2 struts, answering a question of Roth and Whiteley (Trans Am Math Soc 265:419–446, 1981) and solving an open problem of Connelly (Recent progress in rigidity theory, 2008).     

Some results of infinitesimal II-isometry of closed surfaces

In this paper, we continue the discussion of the conjecture which says that infinitesimal II-isometry of surface is infinitesimal I-isometry, i.e., infinitesimally rigid. We have some invariants by means of which some integral formulas are worked out. As an application to these integral formulas, we get some results on infinitesimal II-isometry of closed surface. The theorems proved are just more or less obvious generalizations of known results.     

Global Rigidity: The Effect of Coning

Recent results have confirmed that the global rigidity of bar-and-joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions. Although it is not known if there is a deterministic algorithm that runs in polynomial time and space, to decide if a graph is generically globally rigid, there is an algorithm (Gortler et al. in Characterizing generic global rigidity, arXiv:, 2007) running in polynomial time and space that will decide with no false positives and only has false negatives with low probability. When there is a framework that is infinitesimally rigid with a stress matrix of maximal rank, we describe it as a certificate which guarantees that the graph is generically globally rigid, although this framework, itself, may not be globally rigid. We present a set of examples which clarify a number of aspects of global rigidity.     

Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.     

Reprint of: Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.     

The Orbit Rigidity Matrix of a Symmetric Framework

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions.     

On the infinitesimal affine rigidity of ellipsoids in the n-space

Due to R. Schneider 1967 an ellipsoid E in the affine space \Bbb Aⁿ\Bbb A^n is affinely rigid, i.e. every other ovaloid F in \Bbb Aⁿ\Bbb A^n with the same affine Blaschke metric as for E equals E up to an equiaffine motion of E. Due to M. Kozlowski 1985 resp. W. Blaschke 1922 for n = 3 ellipsoids are moreover S-rigid resp. infinitesimally S-rigid in the sense of equal resp. infinitesimally equal affine scalar curvature S (unknown until now for n >3). - In this article it is proved that ellipsoids in \Bbb Aⁿ\Bbb A^n are also infinitesimally S-rigid for any n.     

Pin-Collinear Body-and-Pin Frameworks and the Molecular Conjecture

T.-S. Tay and W. Whiteley independently characterized the multigraphs which can be realized as an infinitesimally rigid d-dimensional body-and-hinge framework. In 1984 they jointly conjectured that each graph in this family can be realized as an infinitesimally rigid framework with the additional property that the hinges incident to each body lie in a common hyperplane. This conjecture has become known as the Molecular Conjecture because of its implication for the rigidity of molecules in 3-dimensional space. Whiteley gave a partial solution for the 2-dimensional form of the conjecture in 1989 by showing that it holds for multigraphs G=(V,E) in the family which have the minimum number of edges, i.e. satisfy 2|E|=3|V|−3. In this paper, we give a complete solution for the 2-dimensional version of the Molecular Conjecture. Our proof relies on a new formula for the maximum rank of a pin-collinear body-and-pin realization of a multigraph as a 2-dimensional bar-and-joint framework. This work was supported by an International Joint Project grant of the Royal Society. The research of T. Jordán was also supported by the MTA-ELTE Egerváry Research Group on Combinatorial Optimization and the Hungarian Scientific Research Fund grant no. T49671.     

A Variational Proof of Alexandrov’s Convex Cap Theorem

We give a variational proof of the existence and uniqueness of a convex cap with the given metric on the boundary. The proof uses the concavity of the total scalar curvature functional (also called Hilbert-Einstein functional) on the space of generalized convex caps. As a by-product, we prove that generalized convex caps with the fixed metric on the boundary are globally rigid, that is uniquely determined by their curvatures. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces”.     

Rigid Curves in Complete Intersection Calabi–Yau Threefolds

Working over the complex numbers, we study curves lying in a complete intersection K3 surface contained in a (nodal) complete intersection Calabi–Yau threefold. Under certain generality assumptions, we show that the linear system of curves in the surface is a connected componend of the the Hilbert scheme of the threefold. In the case of genus one, we deduce the existence of infinitesimally rigid embeddings of elliptic curves of arbitrary degree in the general complete intersection Calabi–Yau threefold.     

Upper Semicontinuous Valuations on the Space of Convex Discs

We show that every rigid motion invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, the length, the area, and a suitable curvature integral of the convex disc.     

具有不可缩小Beltrami系数的拟共形映射

本文讨论了不可缩小Beltrami系数与无限小不可缩小Beltrami系数的局部 性与整体性的关系;并通过烟囱形区域上仿射拉伸的实例,指出构造一类极值的具有 非常数模的不可缩小Beltrami系数与无限小不可缩小Beltrami系数的方法.     

Infinitesimally flexible meshes and discrete minimal surfaces

We explore the geometry of isothermic meshes, conical meshes, and asymptotic meshes around the Christoffel dual construction of a discrete minimal surface. We present a discrete Legendre transform which realizes discrete minimal surfaces as conical meshes. Conical meshes turn out to be infinitesimally flexible if and only if their spherical image is isothermic, which implies that discrete minimal surfaces constructed in this way are infinitesimally flexible, and therefore possess reciprocal-parallel meshes. These are discrete minimal surfaces in their own right. In our study of relative kinematics of infinitesimally flexible meshes, we encounter characterizations of flexibility and isothermicity which are of incidence-geometric nature and are related to the classical Desargues configuration. The Lelieuvre formula for asymptotic meshes leads to another characterization of isothermic meshes in the sphere which is based on triangle areas.     

Wrapping spheres with flat paper

We study wrappings of smooth (convex) surfaces by a flat piece of paper or foil. Such wrappings differ from standard mathematical origami because they require infinitely many infinitesimally small folds (“crumpling”) in order to transform the flat sheet into a surface of nonzero curvature. Our goal is to find shapes that wrap a given surface, have small area and small perimeter (for efficient material usage), and tile the plane (for efficient mass production). Our results focus on the case of wrapping a sphere. We characterize the smallest square that wraps the unit sphere, show that a 0.1% smaller equilateral triangle suffices, and find a 20% smaller shape contained in the equilateral triangle that still tiles the plane and has small perimeter.     

Über Deformationen von analytischen Abbildungskeimen

The notion of deformations of germs of k-analytic mappings generalizes the one of deformations of germs of k-analytic spaces. Using algebraic terms, we prove:

1. The morphism f: A→B of analytic algebras is rigid, iff it is infinitesimally rigid. Moreover, this is equivalent to Ex_A (B,B)=0. This theorem generalizes a result of SCHUSTER [11].
2. Let A be a regular analytic algebra. Then f is rigid iff there exists a rigid analytic algebra B_o such that f is equivalent to the canonic injection A→A?B_o.
3. If f is “almost everywhere” rigid or smooth, then the injection Ext _B ^l (Ω_B|A, Bⁿ)→Ex_A(B, Bⁿ) is an isomorphism.

     

(r,s)-STABILITY OF UNFOLDING OF Γ-EQUIVARIANT BIFURCATION PROBLEM

In this paper,the Γ-equivariant (s, t)-equivalence relation and Γ-equivariant infinitesimally (r, s)-stability of Γ-equivariant bifurcation problem are defined. The criterion for Γ-equivariant infinitesimally (r, s)-stability is proven when Γ is a compact finite Lie group . Transversality condition is used to characterize the stability.