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Elimination of parasitic solutions in the theory of flexible polyhedra. (English) Zbl 1469.52018

Summary: The action of the rotation group SO(3) on systems of \(n\) points in the 3-dimensional Euclidean space \(\mathbf{R}^3\) induces naturally an action of SO(3) on \(\mathbf{R}^{3n}\). In the present paper we consider the following question: do there exist 3 polynomial functions \(f_1,f_2,f_3\) on \(\mathbf{R}^{3n}\) such that the intersection of the set of common zeros of \(f_1,f_2\), and \(f_3\) with each orbit of SO(3) in \(\mathbf{R}^{3n}\) is nonempty and finite? Questions of this kind arise when one is interested in relative motions of a given set of \(n\) points, i. e., when one wants to exclude the local motions of the system of points as a rigid body. An example is the problem of deciding whether a given polyhedron is non-trivially flexible. We prove that such functions do exist. To get a necessary system of equations \(f_1=0,f_2=0,f_3=0\), we show how starting by choice of a hypersurface in \(\mathbf{CP}^{n-1}\) containing no conics, no lines, and no real points one can find such a system.

MSC:

52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
70B15 Kinematics of mechanisms and robots
51F25 Orthogonal and unitary groups in metric geometry
52B10 Three-dimensional polytopes
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