Buffon投针问题解的几何解释及其在球面上的推广

古典Buffon投钎问题是由Buffon于1777年提出并解决的。自那以来,人们对Buffon问题作了种种推广,但考虑的都是随机投入的物体与等距离平行直线或平行超平面相交的概率。本文将对短针问题作如下的推广:1,在E_2(E_3)中,K_4(i=1,2,…)是以O为中心,id为半径的圆盘(球体),则长为l(≤d)的随机短针与 K_ 相交的概率等于2l/(πd)(l/(2d)),从而给出了Buffon问题解的新的几何解释;2,在单位球面S_2上画n-1个距离为d=π/n的纬线圆,导出了S~2上的随机短针与这些纬线圆相交的概率公式和极限结果,同时还导出了随机小球冠与纬线圆相交的概率公式和极限结果,这些极限结果的形式极为简单,且具有明确的几何意义。

A Geometric Interpretation of the Buffon Needle Problem and Its Extension to a Sphere

The classical Buffon needle problem was forward and solved by Buffon in 1777. Since then,mathematicians have done various extensions for Buffon problem.Their considerations were to look for the probability when an object dropped at random would intersect one of parallel lines or parallel hyperplanes.This paper extends Buffon short needle problem.A new geometric interpretation of Buffon needle problem has been given.Also,this paper gives the probability formula and its limit result that not only a short needle but a random small cap dropped at random intersects one of the parallels on a unit sphere.The rcsults are proven very simple in form but explicit in geometric sense.