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arXiv · 2002.03995

Elimination of parasitic solutions in theory of flexible polyhedra

I. Kh. Sabitov·D. A. Stepanov
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Abstract

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 $R^{3n}$ 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.

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BibTeXRIS

I. Kh. Sabitov, D. A. Stepanov. 2020-02-10. Elimination of parasitic solutions in theory of flexible polyhedra. https://arxiv.org/abs/2002.03995

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