具有随机分布任意闭轨线的域的平均弧定理
Mean arc theorem for exploring domains with randomly distributed arbitrary closed trajectories
摘要：
积分几何的一个显着结果是柯西公式，它将随机穿过凸2D域的弹道轨迹的平均路径长度与区域面积及其周长之间的比率联系起来。这个定理已被推广到非凸区域，并被推广到布朗运动的情形，在生物运动和波动物理等领域有着广泛的应用。在这里，我们推广的定理，探索任意域的任意封闭的轨迹。我们证明，无论轨迹的复杂性，平均弧长仍然满足柯西公式，提供没有轨迹是完全包含在域中。低于该阈值，平均弧长随着轨迹的大小而减小。在这种情况下，对于与凸域相交的凸轨线，只要它们比较小，仍然可以给出近似解析公式。为了验证我们的分析，我们进行了不同类型的轨迹探索任意二维域的数值模拟。我们的结果可以应用于有界域的几何信息检索的平均第一个入口-出口长度。
英文摘要：
A remarkable result from integral geometry is Cauchy's formula, which relates the mean path length of ballistic trajectories randomly crossing a convex 2D domain, to the ratio between the region area and its perimeter. This theorem has been generalized for non-convex domains and extended to the case of Brownian motion to find many applications in various fields including biological locomotion and wave physics. Here, we generalize the theorem to arbitrary closed trajectories exploring arbitrary domains. We demonstrate that, regardless of the complexity of the trajectory, the mean arc length still satisfies Cauchy's formula provided that no trajectory is entirely contained in the domain. Below this threshold, the mean arc length decreases with the size of the trajectory. In this case, an approximate analytical formula can still be given for convex trajectories intersecting convex domains provided they are small in comparison. To validate our analysis, we performed numerical simulations of different types of trajectories exploring arbitrary 2D domains. Our results could be applied to retrieve geometric information of bounded domains from the mean first entrance-exit length.
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