Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 45, No. 1, pp. 1320 (2004) 

On Buffon's problem for a lattice and its deformationsGiuseppe Caristi and Massimiliano FerraraDepartment of Economic and business branches of knowledge, Faculty of Economics, University of Messina, Via dei Verdi, 75  98121 Messina, Italy; email: gcaristi@dipmat.unime.it email: massiferrara@tiscalinet.itAbstract: We consider the Buffon's problem for the lattice $R_{\alpha,a}$ which has the fundamental cell composed by the union of octagon, with all sides of lengths $a$ and the angles $(\pi\alpha)$ and $(\frac{\pi}{2}+\alpha)$ with $\alpha\in\left.\right]0,\frac{\pi}{2}\left[\right.$, and of the square with side of length $a$ (see Fig. 1). We determine the probability of intersection of a body test needle of length $l,\ l<a$. For $\alpha=\frac{\pi}{4}$ we also give the estimate of this probability for the cases, when the segment is nonsmall with respect to $R_{\frac{\pi}{4},a}$ (see [D1], [D2]). [D1] Duma, A.: Problems of Buffon type for ``nonsmall'' needles. Rend. Circ. Mat. Palermo, Serie II, Tomo XLVIII (1999), 2340. [D2] Duma, A.: Problems of Buffon type for ``nonsmall'' needles (II). Rev. Roum. Math. Pures Appl., Tome XLIII, 12 (1998), 121135. Keywords: geometric probability, stochastic geometry, random sets, random convex sets and integral geometry Classification (MSC2000): 60D05, 52A22 Full text of the article:
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