任意多边形的随机距离：两随机点间的数学方法
Random Distances Associated with Arbitrary Polygons: An Algorithmic Approach between Two Random Points
摘要：
本文提出了一种新的算法方法来计算任意多边形中均匀分布的两点之间的距离分布，该方法基于积分几何中的扩展运动学测度（KM）。我们首先获得与任意三角形相关联的这种随机点距离分布（PDD）（即，三角形PDD），包括三角形内的PDD，以及共享公共边或公共顶点的两个三角形之间的PDD。对于每一种情况下，我们提供了一个算法程序显示的数学推导过程中，基于该封闭形式的表达式或算法的结果可以得到。所得到的三角形PDD可以用于建模和分析与三角形几何形状相关联的无线通信网络，例如具有三角形簇的传感器网络和具有高度定向天线的三角形蜂窝系统。此外，基于所获得的三角形PDD，我们然后展示了如何通过分解和递归的方法来获得与任意多边形相关联的PDD，因为任何多边形都可以被三角化，并且任何几何形状都可以用多边形以所需的精度来近似。最后，我们给出了与环几何相关的PDD。本文的研究结果丰富和拓展了概率距离模型在无线通信网络分析中的理论和应用。
英文摘要：
This report presents a new, algorithmic approach to the distributions of the distance between two points distributed uniformly at random in various polygons, based on the extended Kinematic Measure (KM) from integral geometry. We first obtain such random Point Distance Distributions (PDDs) associated with arbitrary triangles (i.e., triangle-PDDs), including the PDD within a triangle, and that between two triangles sharing either a common side or a common vertex. For each case, we provide an algorithmic procedure showing the mathematical derivation process, based on which either the closed-form expressions or the algorithmic results can be obtained. The obtained triangle-PDDs can be utilized for modeling and analyzing the wireless communication networks associated with triangle geometries, such as sensor networks with triangle-shaped clusters and triangle-shaped cellular systems with highly directional antennas. Furthermore, based on the obtained triangle-PDDs, we then show how to obtain the PDDs associated with arbitrary polygons through the decomposition and recursion approach, since any polygons can be triangulated, and any geometry shapes can be approximated by polygons with a needed precision. Finally, we give the PDDs associated with ring geometries. The results shown in this report can enrich and expand the theory and application of the probabilistic distance models for the analysis of wireless communication networks.
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