论文标题
移动图的耦合器曲线和计算刚性图的实现
Coupler curves of moving graphs and counting realizations of rigid graphs
论文作者
论文摘要
书法是一张图表，对于几乎所有边缘长度分配，如果我们固定边缘并将顶点视为旋转的接头，则在平面上以一种自由度移动。杰出的书法顶点的轨迹称为其耦合器曲线。对于每个书法，我们唯一地分配了由三个整数组成的矢量。该矢量界定了耦合器曲线不可还原成分的程度和几何属。旋转和翻译的图表有限很多，但至少有两个实现在平面上几乎所有边缘长度分配，都是两个书法的结合。我们表明，这一数量的实现等于与这两个书法相关的向量的某些内部产物。作为一个应用程序，我们获得了一种改进的算法，以计算实现数量的数量，并通过计算实现，我们表征了耦合器曲线的不变性。
A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. The trajectory of a distinguished vertex of the calligraph is called its coupler curve. To each calligraph we uniquely assign a vector consisting of three integers. This vector bounds the degrees and geometric genera of irreducible components of the coupler curve. A graph, that up to rotations and translations admits finitely many, but at least two, realizations into the plane for almost all edge length assignments, is a union of two calligraphs. We show that this number of realizations is equal to a certain inner product of the vectors associated to these two calligraphs. As an application we obtain an improved algorithm for counting numbers of realizations, and by counting realizations we characterize invariants of coupler curves.
