Buffon's Problem With Convex Domain In Rectangle Lattices

Integral Geometry and Geometric Probability are originated from the well-known Buffon Problem in1733. From then on, Crofton,Czuer,Poincare,Ddlthei had made great contributions respectively. For many years Santalo was the father figure in Integral Geometry, and related research work had done by Chen Xinshen,Yan Zhida,Wu Daren,Ren Deling in china.Buffon had discussed the famous Buffon needle problem in the appendix of his research report in 1733. Then the extended applications of Buffon problem have been discussed quite often Santalo had extended the needle to a convex domain[12], (the diameter of the convex domain is less than the distance of the grid of parallel lines) .Ren Delin had extended it to long needle problem, who cancelled the limitation (the diameter of the convex domain is less than the distance of the grid of parallel lines).In this paper, the numbers of the distributions of convex domain and two linear orthogonal parallel lines will be discussed on the basis of the Buffon's Problem with convex domain in rectangle lattices achieved by former researchers.