论文标题
消除柔性多面体理论中的寄生解决方案
Elimination of parasitic solutions in theory of flexible polyhedra
论文作者
论文摘要
旋转组$ so（3）$在$ n $点的系统中的动作在$ 3 $二维的Euclidean Space $ \ Mathbf {r}^3 $自然会诱导$ SO（3）$ on $ \ MATHBF {r}^r}^{3nn} $的动作。在本文中，我们考虑以下问题：是否存在$ 3 $多项式函数$ f_1 $，$ f_2 $，$ f_3 $，$ \ m \ m i \ mathbf {r}^{3n} $，以使得$ f_1 $ f_1 $，$ f_2 $的$ f_1 $和$ f_3 $的共同零的相交有限？当人们对给定的$ n $点的相对动作感兴趣时，即当一个人想将点系统的本地动作作为刚体时，就会出现这种问题。一个例子是确定给定的多面体在非平整灵活性上是否是一个问题。我们证明确实存在这样的功能。要获得一个必要的方程系统$ f_1 = 0 $，$ f_2 = 0 $，$ f_3 = 0 $，我们显示如何通过选择$ \ mathbf {cp}^{n-1} $中的hypersurface开始，其中包含圆锥，没有线条，没有线条，没有真实的点可以找到这样的系统。
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.
