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Flexible octahedra in the projective extension of the Euclidean 3-space. (English) Zbl 1235.52032

A polyhedron (i.e., a polyhedral surface) is said to be flexible if its spatial shape can be changed continuously due to changes of its dihedral angles only (i.e., in such a way that every face remains congruent to itself during the flex). The study of the flexible polyhedra in Euclidean spaces has a long history that starts with the paper by R. Bricard [Journ. de Math. (5) 3, 113–148 (1897; JFM 28.0624.01)] who found flexible octahedra. An account of the modern state of this theory may be found in [I. Kh. Sabitov, Russ. Math. Surv. 66, No. 3, 445–505 (2011; Zbl 1230.52031)].
The author studies flexible octahedra in the projective extension of the Euclidean 3-space. If all vertices are finite, every flexible octahedron is a Bricard octahedron. All flexible octahedra with one vertex at infinity were described in the author’s paper “Self-motions of TSSM manipulators with two parallel rotary axes” [submitted to the ASME Journal of Mechanisms and Robotics]. In the paper under review, the author finds all types of flexible octahedra with two vertices at infinity and proves that all flexible octahedra with at least three vertices at infinity are trivially flexible. This completes the classification of flexible octahedra in the projective extension of the Euclidean 3-space.
The author approaches the problem from the kinematical point of view. In particular, he uses recent results on Kokotsakis meshes and reducible compositions of two four-bar linkages.

MSC:

52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
53A17 Differential geometric aspects in kinematics
52B10 Three-dimensional polytopes
70B15 Kinematics of mechanisms and robots
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