Projective background of the infinitesimal rigidity of frameworks

We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the second one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality between statics and kinematics is established through the principles of virtual work. A geometric approach to statics, due essentially to Grassmann, makes both theorems straightforward. Besides, it provides a simple derivation of the formulas both for the Darboux-Sauer correspondence and for the infinitesimal Pogorelov maps.

[1]  R. Stanley.  The number of faces of a simplicial convex polytope , 1980 .

[2]  G. Thomsen.  Uber affine Geometrie XXXIX , 1923 .

[3]  R. Connelly.  In Handbook of Convex Geometry , 1993 .

[4]  Gil Kalai,et al.  Rigidity and the lower bound theorem 1 , 1987 .

[5]  W. Blaschke.  Über affine Geometrie XXVI1): Wackelige Achtflache , 1920 .

[6]  W. Whiteley,et al.  Statics of Frameworks and Motions of Panel Structures: A projective Geometric Introduction , 1982 .

[7]  Walter Whiteley,et al.  SOME NOTES ON THE EQUIVALENCE OF FIRST-ORDER RIGIDITY IN VARIOUS GEOMETRIES , 2007, 0709.3354.

[8]  Hyperbolic manifolds with convex boundary , 2002, math/0205305.

[9]  T. Braden.  Remarks on the Combinatorial Intersection Cohomology of Fans , 2005, math/0511488.

[10]  H. Gluck.  Almost all simply connected closed surfaces are rigid , 1975 .

[11]  Neil White,et al.  Skeletal rigidity of simplicial complexes, II , 1995, Eur. J. Comb..

[12]  Neil White,et al.  Skeletal rigidity of simplicial complexes, I , 1995, Eur. J. Comb..

[13]  W. Whiteley.  INFINITESIMAL MOTIONS OF A BIPARTITE FRAMEWORK , 1984 .

[14]  Peter McMullen,et al.  Weights on polytopes , 1996, Discret. Comput. Geom..

[15]  Jean-Marc Schlenker.  A Rigidity Criterion for Non-Convex Polyhedra , 2005, Discret. Comput. Geom..

[16]  W. Whiteley.  Infinitesimally rigid polyhedra. I. Statics of frameworks , 1984 .

[17]  Projektive Sätze in der Statik des starren Körpers , 1935 .

[18]  W. Whiteley.  Motions and stresses of projected polyhedra , 1982 .

[19]  R. Connelly.  The rigidity of certain cabled frameworks and the second-order rigidity of arbitrarily triangulated convex surfaces , 1980 .

[20]  E. Bolker,et al.  When is a bipartite graph a rigid framework , 1980 .

[21]  Peter McMullen,et al.  On simple polytopes , 1993 .

[22]  A. V. Pogorelov.  Extrinsic geometry of convex surfaces , 1973 .

[23]  Walter Whiteley,et al.  Rigidity and polarity , 1987 .

[24]  M. Dehn,et al.  Über die Starrheit konvexer Polyeder , 1916 .

[25]  Polyhedral hyperbolic metrics on surfaces , 2008, 0801.0538.

[26]  Carl W. Lee,et al.  P.L.-Spheres, convex polytopes, and stress , 1996, Discret. Comput. Geom..

[27]  A. Cauchy.  Oeuvres complètes: Sur les polygones et les polyèdres (Second Mémoire) , 2009 .

[28]  G. Darboux.  Leçons sur la théorie générale des surfaces , 1887 .

[29]  Felix Klein.  Elementary Mathematics from an Advanced Standpoint: Geometry , 1941 .

[30]  Walter Whiteley,et al.  A Homological Interpretation of Skeletal Rigidity , 2000, Adv. Appl. Math..

[31]  LVI. On the application of Barycentric perspective to the transformation of structures , 1863 .

[32]  Jörg M. Wills,et al.  Handbook of Convex Geometry , 1993 .

[33]  P. Zsombor-Murray,et al.  Elementary Mathematics from an Advanced Standpoint , 1940, Nature.

[34]  Walter Wunderlich,et al.  Projective invariance of shaky structures , 1982 .

[35]  R. Connelly.  CHAPTER 1.7 – Rigidity , 1993 .