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Projective background of the infinitesimal rigidity of frameworks

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Abstract

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.

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Authors and Affiliations

  1. Institut für Mathematik, MA 8-3, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Ivan Izmestiev

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  1. Ivan Izmestiev
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Correspondence to Ivan Izmestiev.

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The research for this article was supported by the DFG Research Unit “Polyhedral Surfaces”.

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Izmestiev, I. Projective background of the infinitesimal rigidity of frameworks. Geom Dedicata 140, 183–203 (2009). https://doi.org/10.1007/s10711-008-9339-9

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