Buffon-Laplace Type Problem for an Irregular Lattice

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1 Applied Mathematical Sciences Vol no HIKARI Ltd Buffon-Laplace Type Problem for an Irregular Lattice Ersilia Saitta Department of Economics University of Messina Via dei Verdi Messina Italy Marius Stoka Sciences Academy of Turin Via Maria Vittoria Torino Italy Copyright c 16 Ersilia Saitta and Marius Stoka. This article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Abstract In this paper we consider a lattice with the fundamental cell represented in fig.1 and we compute the probability that a segment of random position and of constant length intersects a side of the lattice. Mathematics Subject Classification: 6D5 5A Keywords: Geometric Probability stochastic geometry random sets random convex sets and integral geometry 1 Introduction In recent years the Buffon-Laplace type problems for irregular lattices have been studied by several authors [1] [] [3] [4] [5] [6] and [7]. They have considered classical problems with obstacles and with maximum probability Buffon-Laplace type problems. In particular Caristi and Stoka [8] have considered a Laplace type problem for a regular lattice but with an irregular cell.

2 73 Ersilia Saitta and Marius Stoka Moreover considering the fundamental results obtained by Poincaré [9] Stoka [1] we want to consider another particular irregular lattice composed by two isoscele triangles and two isoscele trapezium showing the geometric point of view also. We computed the probability that a random segment of constant length intersects the fundamental cell. Main Result Let R (a b; ) be the lattice with fundamental cell C represented in fig. 1 A b D C 3 F a C 1 C E C 4 B C fig.1 where ] π ] and a b. 4 From this figure we have BAF = ÂBE = π ÂF E = BEF = π + (1) b BE = CE = AF = DF = EF = a btg. () cos We want to compute the probability that a segment s with a random position and of constant length l < a i.e. the probability P int that the segment s intersects a side of the fundamental cell C. We denoted by ϕ [ π ] the angle that the segment s formed with the line BE. The position of the segment s is determinated by its centre and by the angle ϕ defined. To compute the probability P int we considered the limiting positions of segment s for a specified value of ϕ in the cells C i. Thus we have fig.

3 Buffon-Laplace type problem for an irregular lattice 733 A A c 1 A 5 A 4 c 5 D D 1 c 4 D A 1 a 1 A 3 A 6 a 6 C! 3(ϕ) c c 3 F b 4 F F3 b 3 D 3 D 4 F 5 F 1 F 4 a C! 1(ϕ) a 5 b 5 C! (ϕ) b E 4 E 5 E 1 a 4 E a 3 E E3 b 6 C 4 B 1 B 4 C 1 C! 4(ϕ) b 1 B ϕ ϕ B 3 ϕ B fig. C C 3 C 5 C and the following: areaĉ1 (ϕ) = areaĉ (ϕ) = areac 1 areaĉ3 (ϕ) = areaĉ4 (ϕ) = areac 3 From fig. we have: 6 areaa i (ϕ) (3) 5 areac i (ϕ). (4) areaa 1 (ϕ) = l cos (ϕ + ) sin (ϕ + ) cos areaa (ϕ) = al cos (ϕ + ) l cos (ϕ + ) sin (ϕ + ) cos areaa 4 (ϕ) = l sin ϕ cos (ϕ + ) cos areaa 3 (ϕ) = bl sin ϕ l sin ϕ cos (ϕ + ) cos areaa 5 (ϕ) = a btg areaa 6 = l cos (ϕ + ) l sin ϕ cos (ϕ + ) cos bl l sin (ϕ + ) cos (ϕ + ) sin (ϕ + ).

4 734 Ersilia Saitta and Marius Stoka Replacing in (3) we have: where areaĉ1 (ϕ) = areaĉ (ϕ) = areac 1 A 1 (ϕ) (5) A 1 (ϕ) = 6 areaa i (ϕ) = al cos (ϕ + ) + For areaĉ3 we have: l cos (ϕ + ) bl sin ϕ cos [sin (ϕ + ) + sin ϕ]. (6) areac 1 (ϕ) = l sin (ϕ + ) sin (ϕ + ) sin areac 4 (ϕ) = l sin ϕ sin (ϕ ) sin areac 5 (ϕ) = bl sin (ϕ + ) l sin (ϕ + ) sin [sin (ϕ + ) + sin (ϕ )] areac (ϕ) = bl sin (ϕ + ) l sin (ϕ + ) sin (ϕ + ) sin bl sin (ϕ ) areac 3 (ϕ) = Replacing in (4) we have: l sin ϕ sin (ϕ ). sin where areaĉ3 (ϕ) = areaĉ4 (ϕ) = areac 3 A 3 (ϕ) (7) A 3 (ϕ) = 5 areac i (ϕ) = bl [sin (ϕ + ) + sin (ϕ )] + bl sin (ϕ + ) l sin (ϕ + ) [sin (ϕ + ) + sin (ϕ )]. (8) sin Using formulas (6) and (8) we obtain A 1 (ϕ) + A 3 (ϕ) = al cos (ϕ + ) + bl sin (ϕ + ) + bl

5 Buffon-Laplace type problem for an irregular lattice 735 [sin (ϕ + ) + sin (ϕ ) + sin ϕ] l sin (ϕ + ) sin [sin (ϕ + ) + sin (ϕ )] l cos (ϕ + ) [sin (ϕ + ) + sin ϕ]. (9) Denoting by M 1 and M 3 the set of the segments s which have their centre in C 1 respectly C 3 and with N 1 and N 3 the set of segments s all contained in the cell C 1 respectively C 3 we have (cf. [1]): P int = 1 µ (N 1) + µ (N 3 ) µ (M 1 ) + µ (M 3 ) (1) where µ is the Lebesgue measure in the Euclidean plane. To compute the above measure µ (M 1 ) µ (M 3 ) µ (N 1 ) and µ (N 3 ) we used the Poincaré kinematic measure [9]: dk = dx dy where x y are the coordinates of the centre of s and ϕ the fixed angle. From fig. we have: ϕ π. then We have µ (M 1 ) = µ (M 3 ) = (areac 1 ) = (areac 3 ) = dxdy = {(xy)ɛc 1 } ) areac 1 dxdy = {(xy)ɛc 3 } ) areac 3 ) ab µ (M 1 ) + µ (M 3 ) =. (11) For the same reason considering (5) (6) (7) and (8) we can write

6 736 Ersilia Saitta and Marius Stoka then µ (N 1 ) = µ (N 3 ) = {(xy)ɛĉ1} dxdy = ) areac 1 {(xy)ɛĉ3} dxdy = [areac 3 A 3 (ϕ)] = [( )] areaĉ1 (ϕ) = [A 1 (ϕ)] [( )] areaĉ3 (ϕ) = ) areac 3 [A 3 (ϕ)] ) ab µ (N 1 ) + µ (N 3 ) = [A 1 (ϕ) + A 3 (ϕ)]. (1) The (1) (11) and (1) give us P int = And by (9) we obtain: l 4 [ Consequently [ 4 [A 1 (ϕ) + A 3 (ϕ)]. (π ) ab [A 1 (ϕ) + A (ϕ)] = al (1 sin ) + bl (sin sin + cos + cos ) + bl cos + 4 cos + cos cos + ) (cos + cos ) sin P int = { 4 al (1 sin ) + (π ) ab bl cos + bl (sin sin + cos + cos ) + l 4 4 cos + cos cos + ) (cos + cos ) sin ] ]}..

7 Buffon-Laplace type problem for an irregular lattice 737 References [1] U. Baesel Geometric probabilities for a cluster of needles and a lattice of parallel planes Supp. Rend. Circ. Mat. Di Palermo (11) no [] V. Bonanzinga Buffon s problem with a 3-star and a lattice of parallelograms Supp. Rend. Circ. Mat. Di Palermo (1) no [3] D. Barilla G. Caristi M. Stoka Laplace Type Problem for an Irregular Lattice with Cell Composed by Two Isoscele Triangles and an Isoscele Trapezium International Journal of Mathematical Analysis 9 (15) no [4] D. Barilla G. Caristi M. Stoka Laplace type problem for an irregular lattice International Journal Contemporary Mathematical Sciences 1 (15) no [5] D. Barilla G. Caristi A. Puglisi A Buffon - Laplace type problems for an irregular lattice and with maximum probability Applied Mathematical Sciences 8 (14) no [6] G. Caristi M. Stoka A Buffon - Laplace type problem for an irregular lattice and body test rectangle Applied Mathematical Sciences 8 (14) no [7] G. Caristi M. Stoka A Buffon - Laplace type problem for an irregular lattice and different obstacles Applied Mathematical Sciences 9 (15) no [8] G. Caristi M. Stoka A Laplace type problem for regular lattices with irregular hexagonal cell Far East Journal Mathematical Sciences 5 (11) no [9] H. Poincaré Calcul des Probabilités ed. Gauthier Villars Paris 191. [1] M. Stoka Probabilités géométriques de type Buffon dans le plan euclidien Atti Acc. Sci. Torino 11 ( ) Received: June 15 16; Published: March 14 17