星形平面域中相交随机线段的角度分布
Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains
中文总结 AI 辅助
针对星形平面域中均匀随机点构成的相交线段，推导了其夹角条件分布的积分表示，揭示了边界几何对角度分布的影响，并关联Sylvester四点问题。
中文摘要 AI 辅助
设 $\Omega\subset\mathbb{R}^2$ 为关于原点星形的有界平面集，且 $A,B,C,D$ 为在 $\Omega$ 上均匀分布的独立随机点。我们考虑随机线段 $S_{AB}$ 和 $S_{CD}$，并研究在它们相交的条件下，两者所形成较小角 $\Theta\in[0,\pi/2]$ 的分布。利用 $\Omega$ 的径向函数，以及每条线段以其支撑线和端点在该线上的位置为参数的参数化方法，我们推导出条件分布 \\[ \Pr\{\Theta\leq\theta | S_{AB}\cap S_{CD}\neq\varnothing\} \\] 的积分表示。所得表达式明确展示了 $\Omega$ 边界的几何形状如何决定角度分布。相交概率自然地作为归一化常数出现，并与 Sylvester 四点问题的概率版本相关。
英文摘要
Let $Ω\subset\mathbb{R}^2$ be a bounded planar set that is star-shaped with respect to the origin, and let $A,B,C,D$ be independent random points uniformly distributed on $Ω$. We consider the random segments $S_{AB}$ and $S_{CD}$ and study the distribution of the smaller angle $Θ\in[0,π/2]$ formed by them, conditional on the event that they intersect. Using the radial function of $Ω$, together with a parametrization of each segment in terms of its supporting line and the positions of its endpoints along that line, we derive an integral representation for the conditional distribution \[ \Pr\{Θ\leqθ| S_{AB}\cap S_{CD}\neq\varnothing\}. \] The resulting expression makes explicit how the geometry of the boundary of $Ω$ determines the angular distribution. The probability of intersection appears naturally as the normalizing constant and is related to the probabilistic version of Sylvester's four-point problem.
