The probability of covering a sphere with N circular caps
The probability of covering a sphere with N circular caps
复制标题
用 N 个圆形帽覆盖球体的概率
DOI：
10.1093/biomet/52.3-4.323
复制
发表时间：
1965
期刊：
影响因子：
2.7
通讯作者：
E. Gilbert
中科院分区：
文献类型：
--
作者：
E. Gilbert
瓶盖C，，...，如果单位球面上的每个点x至少属于一个帽，则CQn覆盖球面。X1，...，XN是在球体表面上以恒定概率密度1/（47 r）随机独立选择的。设P（N）为N个帽覆盖球面的概率。本文将部分从理论上和部分通过计算机模拟实验来估计P（N）。在p = 90 °的特殊情况下，精确公式P（N）= 1（N2-N+2）.2-N成立. Kendall & Moran（1963）提到P（N）是几何概率中一个困难问题的例子。Moran和Fazekas de St Groth（1962）给出了P（N）的一个近似公式，并在p = 530 26 '时用实验方法估计了P（N）。这个角度出现在一个有趣的生物学应用中，其中球体是病毒X1，...，XN是抗体攻击病毒的点，P（N）是N个抗体使病毒无害的概率。或者，如果随机分布在地球上的N个观测站，每个观测站都能以p或更小的角度观察天顶，那么P（N）是天空中没有任何点对所有N个观测站隐藏的概率。每个帽覆盖一个分数f= sin 2 p（1）
The caps C,, ..., CQn cover the sphere if every point x on the surface of the unit sphere belongs to at least one cap. X1, ..., XN are to be chosen independently at random with constant probability density 1/(47r) over the surface of the sphere. Let P(N) be the probability that the N caps cover the sphere. This note will estimate P(N) partly theoretically and partly by a computer simulation experiment. An exact formula P(N) = 1(N2-N+2).2-N holds in the special case p = 90?. For other values of p only bounds on P(N) are found. Kendall & Moran (1963) mention P(N) as an example of a difficult problem in geometrical probability. Moran & Fazekas de St Groth (1962) give an approximation formula for P(N) and estimate P(N) experimentally for the value p = 530 26'. This angle occurs in an interesting biological application in which the sphere is a virus, X1, ..., XN are points at which antibodies attack the virus, and P(N) is the probability that N antibodies render the virus harmless. Alternatively if N observatories, situated at random on the Earth, can each look at angles p or less away from the zenith then P(N) is the probability that no point in the sky is hidden from all N observatories. Each cap covers a fraction f=sin2 p (1)