The Buffon Problem. Several Special Convex Body

Buffon's problem aims at solving the problem that throw a small needle randomly in the particular lattice which is based on a convex to get the probability of intersection between the needle and the lattice. Professor Delin Ren put the conceptions of restricted chord function and the restricted chord projection function into it and developing the generalized Buffon's problem. The paper refers to the following aspects:(1) The restricted chord function and the restricted chord projection function are used to research the Buffon needle problem in Regular Triangle Cylinder for one kind of Bounded Lattices and establish the individual expressions for the kinematic measure for Regular Triangle Cylinder. Its application to geometric probability problems will lead to a new conclusion for geometric probability.(2)After considering the Buffon's problem for the lattice Rα,a which has the fundamental cell composed by the union of octagon, with all sides of lengths a and the anglesπ-αandπ/2 +αwith0 <α<π/2, and of the square with side of length a . The probability of intersection of a body testneedle of length l are determined, l < a. Forα=π/4 we also give the estimate of this probability for the cases, when the segment is non-small with respect to Rα,a.(3)Study the Buffon's problem that the fundamental cell composed by a right-angled prism, and get the probability of intersection between a random sphere and the sides of R .