标题： Simuorb：一种生成和描述团排列交点的新方法 显示英文标题

标题： Simuorb: a new method for generating and describing the intersection points of clique-arrangements

摘要： 这项工作可视为对\cite{RS2}的补充，处理正多边形对角线形成的交点的分布方式。 最近作者指出，这些点仅位于原点中心的圆上，并且它们的半径分别依赖于最初正$n$-gon 的四个顶点索引，这四个顶点决定了交点背后的两条直线。 由于这四个顶点位于单位圆内接正$n$-gon 上的预设位置，因此可以用 3 个参数来描述轨道，而不是 4 个参数，大致描述前三个顶点之间的路径长度，以及由这四个顶点描述的四边形是简单还是复杂。 这种方法使我们能够处理由团排列生成的轨道，并处理其基数以及相关交点的重数。 为了在不生成相关图的情况下枚举交点，提供了一个基于此三元组策略的可靠计数算法。 在模拟轨道后，我们称这种方法为\textit{模拟轨道}。 该过程稳健、快速，并能全面理解团排列中发生的情况，无论其中包含大量点还是少量点。 显示英文摘要

摘要： This work, which may be seen as a companion paper to \cite{RS2}, handles the way the intersection points made by the diagonals of a regular polygon are distributed. It was stated recently by the authors that these points lie exclusively on circles centered on the origin and also the way their respective radii depend on the four indices of the vertices of the initial regular $n$-gon which characterize the two straight lines underlying the intersection points. Because these four vertices are located at preset positions on the the regular $n$-gon inscribed in the unit circle whose path-length perimeter is constant, it allows the orbits to be characterized by 3 parameters instead of 4, describing roughly the lengths of the paths between the first three vertices, whether the quadrilateral described by these four vertices is simple or complex. This approach enables us to deal with the orbits generated by the clique-arrangement, and to handle their cardinalities as well as the multiplicities of the associated intersection points. A reliable counting-algorithm based on this triplet strategy is provided in order to enumerate the intersection points without generating the associated graph. The orbits being simulated, we call this method \textit{Simuorb}. The procedure is robust, fast and allows a comprehensive understanding of what is happening in a clique-arrangement, whether it contains a large number of points or not.