Study Of The Buffon Probability Based On Plane Grid

The Buffon needle probability is the most classic problem in the field of integral geometry.Santalo has broadened the research of Buffon problem, in which he extended the parallel linelattices to the parallel band lattices. Afterwards, Ren Delin established the system theory for thekinematic measuring of fixed line segment in convex in the two-dimension or n-dimension space,in which he further generalized the research of the Buffon problem. One of the generalizations isbased on the contribution of Santlo, i.e., he adds the limitation condition that the diameter of theconvex domain does not exceed that of the band zone. In addition, he applied the kinematicmeasuring to the geometric probability in the Laplace generalization of the Buffon problem,obtaining the probability of the kinematic measuring.This article researched the generalization of one sort of Buffon problem. First, we havefigured out the kinematic measuring of fixed line segment in some common convexes using thegeneralization support function and the limited chord function, thus the corresponding Buffonprobability. And second, using the kinematic measuring, we obtained the expression of theprobability for a needle intersecting with the bound of complicated lattices, and applied it tosome special case.