A Laplace Type Problem for a Lattice with Cell Composed by Three Triangles with Obstacles

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1 Applied Matheatical Sciences Vol no HIKARI Ltd A Laplace Type Proble for a Lattice with Cell Coposed by Three Triangles with Obstacles Marius Stoka Sciences Acadey of Turin Via Maria Vittoria 101 Torino Italy Copyright c 016 Marius Stoka. This article is distributed under the Creative Coons Attribution License which perits unrestricted use distribution and reproduction in any ediu provided the original work is properly cited. Abstract In this paper a lattice with a cell represented in fig. 1 is considered and we copute the probability that a segent of rando position and of constant length intersects a side of the lattice. Matheatics Subject Classification: 60D05 5A Keywords: Geoetric Probability stochastic geoetry rando sets rando convex sets and integral geoetry 1 Introduction Poincaré [6] and Stoka [7] have obtained the fundaental results for the ost iportant probles of geoetric probles in particular they have solved several Buffon-Laplace type probles. In recent years different authors have considered several Buffon-Lapalce type probles for particular fundaental cells [1] [] [] [] [5] and [6]. Starting fro these results in this paper we consider as fundaental cell a lattice coposed by four isoscele triangles and a rhobus and the Laplace type proble was solved. We copute the probability that a rando segent of constant length intersects the fundaental cell represented in fig. 1.

2 66 Marius Stoka Main Result Let R a ; where is an angle with π π and 0 < a be the lattice with fundaental cell C 0 = C 01 C 0 C 0 represented in fig. 1 A a C 01 C 0 D D D 1 D B B B / / B C 0 C / / C C 1 C fig.1 Fro this figure we obtain the following BC = a cos BE = a cos AE = a sin DE = a cos tg AD = a sin cos tg BD = CD = a cos cos ; 1 B 1 B = B B = C 1 C = C C = sin

3 Three triangles with obstacles 67 D 1 D = D 1 D = D D = sin ; D 1 DD = D 1 DD = D DD = π DD 1 D = DD D 1 = DD 1 D = DD D 1 = DD D = DD D = π 6 ; areabb 1 B = areabb B = areacc 1 C = areacc C = sin areadd 1 D = areadd 1 D = areadd D =. 5 areac 01 = areac 0 = sin a cos cos tg sin 6 areac 0 = a cos tg sin 7 areac 0 = a sin sin. 8 We want to copute the probability that a segent s with a rando position and of constant length l < in a/ a cos intersects a side of lattice R i.e. the probability P int that a segent s intersects a side of the fundaental cell C 0. The position of the segent s is deterinated by its centre and by the angle ϕ fored with the line BC. To copute the probability P int we consider the liiting positions of segent s for a specified value of ϕ in the cells C 0i i = 1. Thus we obtain fig.

4 68 Marius Stoka A a 1 A A 1 c 1 A A 5 A a c ˆ C 01 ϕ a 6 ˆ C 0 ϕ c 6 D 5 D6 D c a 5 B B 6 B B 5 a B / / B 7 B 1 a b 1 D D 8 ˆ C 0 ϕ D D 1 D b 6 D 7 C C 5 C 6 C 7 b ϕ b b B b 5 c C 9 C 8 c 5 C C C 1 C fig. and the following By fig. we have areaĉ01 ϕ = areac 01 areaĉ0 ϕ = areac 0 areaĉ0 ϕ = areac 0 areaa i ϕ 9 areab i ϕ 10 areac i ϕ. 11 areac 5 ϕ = l sin ϕ sin ϕ + sin sin areac ϕ = al cos sin ϕ l sin ϕ l sin ϕ sin sin ϕ +

5 Three triangles with obstacles 69 a cos areac ϕ = cos areac ϕ = l sin ϕ cos ϕ cos. l ϕ sin l sin ϕ cos cos ϕ areac 1 ϕ = l 8 a cos areac 6 ϕ = cos We obtain: ctg + sin ϕ + l ctg 8 cos ϕ l ϕ sin + l sin ϕ sin ϕ + sin. A ϕ = areac i ϕ = al cos sin ϕ l [ cos ctg + 1 In the sae way we have: sin ϕ sin ] + ctg cos ϕ l sin ϕ sin ϕ + sin sin. 1 areaĉ0 ϕ = areac 0 A ϕ. 1 areaa ϕ = l sin cos ϕ areaa 1 ϕ = l sin ϕ cos ϕ cos areaa ϕ = a l sin ϕ l sin ϕ cos ϕ cos areaa 6 ϕ = al sin cos tg cos ϕ l cos ϕ+

6 70 Marius Stoka l sin ϕ tg cos ϕ tg areaa 5 ϕ = l sin ϕ sin ϕ + π 6 a cos areaa ϕ = cos We obtain: A 1 ϕ = [ 1 sin l ϕ sin l sin ϕ sin ϕ + π. 6 areaa i ϕ = al sin cos tg cos ϕ l cos ϕ + cos + cos ] sin ϕ + l sin ϕ tg cos ϕ tg. 1 In the sae way we have: areab ϕ = al areaĉ01 ϕ = areac 01 A 1 ϕ. 15 areab 1 ϕ = l cos ϕ sin ϕ + cos sin cos tg cos ϕ l cos ϕ l cos ϕ sin ϕ + cos areab 5 ϕ = l sin ϕ + sin ϕ + sin areab 6 ϕ = al sin ϕ + l cos ϕ sin ϕ + cos a cos areab ϕ = cos areab ϕ = l sin l ϕ sin + sin l sin ϕ + sin ϕ + sin π sin ϕ l sin ϕ + sin ϕ + sin.

7 Three triangles with obstacles 71 We have: A ϕ = areab i ϕ = al sin cos ϕ + cos sin ϕ + l [ 1 + sin sin cos ϕ + cos ] + sin sin ϕ l [ 1 + cos + sin ctg sin ϕ + tg cos ctg cos ϕ+ tg + cos ctg ] sin. 16 areaĉ0 ϕ = areac 0 A ϕ. 17 Denoting with M i i = 1 the set of segents s which have their centre in C 0i and with N i the set of segents s entirely contained in the cell C 0i we have cf. [10]: P int = 1 µ N i µ M i 18 where µ is the Lebesgue easure in the Euclidean plane. To copute the above easure µ M i and µ N i we used the Poincaré kineatic easure [9]: dk = dx dy dϕ where x y are the coordinates of the centre of s and ϕ the fixed angle. Since [ ϕ π ] µ M i = dϕ dxdy = {xyɛc 0i } areac 0i dϕ = π areac 0i i = 1.

8 7 Marius Stoka and µ N i = µ M i = π areac 0 19 areaĉ0i dϕ = π areac 0i dϕ µ N i = π areac 0 Equations 1 1 and 16 give us [ ] [ A i ϕ dϕ = al 1 sin + cos l {xyɛĉ0i} dxdy = [ ] areaĉ0i A i ϕ dϕ = [A i ϕ] dϕ i = 1 [ ] A 1 ϕ dϕ. 0 sin 1 cos tg + 1 cos sin sin tg + sin + sin tg cos ] [ sin + cos sin cos + sin cos ϕ+ cos + 1 sin sin ϕ ] l [ 1 + cos + sin ctg sin ϕ+ tg ctg cos ctg cos ϕ + tg + ctg + cos ctg ] sin. 1 Forulas and 1 give us the probability P int. For = 0 we have the probability coputed in paper [1].

9 Three triangles with obstacles 7 References [1] U. Baesel A. Dua A Laplace type proble for a lattice of rectangles with triangular obstacles Applied Matheatical Sciences 8 01 no [] D. Barilla M. Bisaia G. Caristi and A. Puglisi On Laplace type probles I Journal of Pure and Applied Matheatics: Advances and Applications no [] D. Barilla M. Bisaia G. Caristi and A. Puglisi On Laplace type probles II Far East Journal of Matheatical Sciences no [] D. Barilla G. Caristi M. Stoka A Buffon-Laplace type proble for a lattice with cell coposed by four triangles and a rhobus with circular section obstacles Applied Matheatical Sciences 8 01 no [5] G. Caristi M. Pettineo M. Stoka A Laplace type proble for three lattices with non-convex cell J. Nonlinear Sci. Appl [6] G. Caristi A. Puglisi M. Stoka A Buffon-Laplace Type Proble for an Irregular Lattice with Cell Coposed by Pentagon + Triangle with Obstacles International Journal of Matheatical Analysis no [7] G. Caristi A. Puglisi M. Stoka A Buffon-Laplace type proble for a lattice with cell coposed by four triangles and a rhobus with triangular obstacles Applied Matheatical Sciences 8 01 no [8] G. Caristi and M. Stoka Soe extension of the Laplace proble Rend. Cric. Mat di Palero [9] H. Poincarè Calcul des Probabilitès nd ed. Gauthier-Villard Paris 191. [10] M. Stoka Probabilités géoétriques de type Buffon dans le plan euclidien Atti Accd. Sci. Torino T Received: July ; Published: January 1 017