http://dx.doi.org/10.12988/ams.2015.5157
A Laplace Type Problem for Irregular Lattice
with Fundamental Cell Composed by Three
Triangles and a Trapetium
D. BarillaDepartment of Economic Science and Quantitative Method University of Messina - Italy
F. Grasso
Department Cognitive Sciences Educational and Cultural Studies University of Messina - Italy
Copyright c 2015 D. Barilla and F. Grasso. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribu-tion, and reproduction in any medium, provided the original work is properly cited.
Abstract
In the previous papers, [1], [2], [3], [4],[5], [6], [7], [8], [9], [10], [11], [12], [13] and [14] the authors studies some Laplace problem for differ-ent lattices. In this paper paper we determine the probability that a random segment of constant length intersects a side of the lattice with fundamental cell represented in fig. 1.
A B C D E F a a a a a a G fig.1
With the rotations of this figure we can write that C0 = C01∪ C02∪ C03∪ C04
and [
ABE = [AEB = \DEF = π
5, \CBE = \DEB = \EDF = \EF D = 2π 5 , \ BCD = \CDE = 3π 5 , BAG = [[ EAG = 3π 4 , (1) |BG| = |EG| = a cosπ 5, |BE| = 2a cos π 5, |AG| = a sin π 5, (2) |DF | = a 2 cosπ5, areaC0 = a2 2 sin π 5 2 cosπ 5 + 1 2 . (3)
We want to compute the probability that a segment s of random position and of constant length l, with l < 4 cosa π
5
, intersects a side of the lattice < (a), i.e. the probability Pint that s intersects a side of the fundamental cell C0.
The position of the segment s is determined by center and by the angle ϕ that it formed with the line CF .
In order to compute the probability Pint we consider the limit positions of
the segment s, for a fixed value of ϕ, in the cells C0i, (i = 1, 2, 3, 4). We have
A B C D E F G A1 A 2 A3 A' A'' A4 B' B'' B1 B2 G ' G 1 G2 G 3 a1 a2 a4 a3 b1 b2 b3 b4 b5 E '' E ' E1 E2 E3 E4 E5 F1 F2 F ' D ' D1 D2 D '' C ' ϕ c1 c2 c3 c 4 c5 c6 d1 d2 d3 d 4 d5 Cˆ03(ϕ ) Cˆ01(ϕ ) Cˆ02(ϕ ) Cˆ04(ϕ ) fig.2 and the relations
area bC01(ϕ) = areaC01− 5 X j=1 areaaj(ϕ) , (4) area bC02(ϕ) = areaC02− 5 X j=1 areabj(ϕ) , (5) area bC03(ϕ) = areaC03− 5 X j=1 areacj(ϕ) , (6) area bC04(ϕ) = areaC04− 5 X j=1 areadj(ϕ) . (7)
By fig.1 and fig.2 we have that: areaa4(ϕ) = l2 4 sin 2ϕ, areaa3(ϕ) = al 2 cos π 5sin ϕ − l2 4 cos ϕ, areaa1(ϕ) = l2cos ϕ sin π5 − ϕ 2 cos π5 ,
areaa2(ϕ) = al 2 sin π 5 − ϕ −l 2cos ϕ sin π 5 − ϕ 2 cosπ5 , areaa5(ϕ) = al 2 sin π 5 cos ϕ − l2 4 sin 2ϕ − l2cos ϕ sin π5 − ϕ 2 cosπ5 . We obtain that: area bC01 = areaC01− A1(ϕ) , where A1(ϕ) = al sin π 5 cos ϕ − l2 4tg π 5 (1 + cos 2ϕ) . In order to compute area bC2(ϕ) we have that:
areab4(ϕ) = l2sin ϕ sin π 5 + ϕ 2 sinπ5 , areab3(ϕ) = al 2 cos π 5sin ϕ − l2sin ϕ sin π 5 + ϕ 2 sinπ5 , areab1(ϕ) = l2cos ϕ sin π 5 + ϕ 2 cosπ5 , areab2(ϕ) = al 2 sin π 5cos ϕ − l2cos ϕ sin π 5 + ϕ cosπ5 , areab5(ϕ) = al 2 sin π 5 + ϕ − l2 2 sin2π5 1 − cos2π 5 cos 2ϕ + sin2π5 2 sin 2ϕ . We obtain that: area bC02 = areaC02− A2(ϕ) , where A2(ϕ) = al sin π 5 + ϕ − l 2 2 sin2π5 1 − cos2π 5 cos 2ϕ + 1 2sin 2π 5 sin 2ϕ .
To compute area bC03(ϕ) we have that
areac4(ϕ) =
l2sin ϕ sin 2π 5 − ϕ
2 sin 2π5 ,
areac3(ϕ) = al 2 sin ϕ − l2sin ϕ sin 2π 5 − ϕ 2 sin2π5 , areac1(ϕ) = l2sin ϕ sin 2π 5 + ϕ 2 sin 2π5 , areac2(ϕ) = al 2 sin 2π 5 + ϕ −l 2sin ϕ sin 2π 5 + ϕ 2 sin2π5 , areac5(ϕ) = al 2 sin 2π 5 − ϕ −l 2sin ϕ sin 2π 5 − ϕ 2 sin2π5 , areac6(ϕ) = al cos π 5sin ϕ − l2sin ϕ sin 2π 5 + ϕ 2 sin2π5 . We obtain that area bC03(ϕ) − A3(ϕ) , where A3(ϕ) = al 2 2 sin2π 5 cos ϕ + 1 + 2 cosπ 5 sin ϕ − l 2 2 sin 2ϕ. To compute now the area bC04(ϕ).
aread4(ϕ) = l2sin ϕ sin 2π 5 + ϕ 2 sin2π5 , aread3(ϕ) = al sin(ϕ) 4 cosπ5 − l sin 2π5 + ϕ sin2π5 , aread1(ϕ) = l2sin 2π 5 − ϕ sin 2π 5 + ϕ 2 sinπ 5 , aread2(ϕ) = al 2 sin 2π 5 − ϕ −l 2sin 2π 5 − ϕ sin 2π 5 + ϕ 2 sinπ5 , aread5(ϕ) = al 2 sin 2π 5 + ϕ − l 2sin 2π 5 − ϕ sin 2π 5 + ϕ 2 sinπ5 − l2sin ϕ sin 2π 5 + ϕ 2 sin2π5 .
We obtain that: area bC04(ϕ) = areaC04− A4(ϕ) , where A4(ϕ) = al sin2π 5 cos ϕ + sin ϕ 4 cos2π5 − l2 4 sin2π5 2 cosπ 5 − cos 2π 5 cos 2ϕ + sin2π 5 sin 2ϕ + 4 cos 2 π 5 + 1 .
Denoting with Mi, (i = 1, 2, 3, 4), the set of segments s that have the center
in the cell C0iand with Ni the set of the segments s completely in C0i, we have
that [16]: Pint = 1 − P4 i=1µ (Ni) P4 i=1µ (Mi) , (8)
where µ is the Lebesgue measure in the euclidean plane.
In order to compute µ (Mi) and µ (Ni) we use the kinematic measure of
Poincar´e [15]:
dK = dx ∧ dy ∧ dϕ,
where x, y are the coordinates of middle point of s and ϕ the fixed angle. We can write: µ (Mi) = π 5 Z 0 dϕ Z Z {(x,y)∈C0i} dxdy = π 5 Z 0 (areaC0i) dϕ = π 5areaC0i, (i = 1, 2, 3, 4) and µ (Ni) = π 5 Z 0 dϕ Z Z {(x,y)∈ bC0i} dxdy = π 5 Z 0 h area bC0i(ϕ) i dϕ = π 5 Z 0 [areaC0i− Ai(ϕ)] dϕ = π 5areaC0i− π 5 Z 0 [Ai(ϕ)] dϕ, (i = 1, 2, 3, 4) .
By these two relations give us:
4 X i=1 µ (Mi) = π 5areaC0, (9)
and 4 X i=1 µ (Ni) = π 5areaC0− π 5 Z 0 " 4 X i=1 Ai(ϕ) # dϕ. (10) follow that: 4 X i=1 Ai(ϕ) = al sinπ 5 2 + 3 cosπ 5 cos ϕ +8 cos 2 π 5 + 2 cos π 5 + 1 4 cosπ5 sin ϕ − l2 4 tgπ 5 − 3ctg 2π 5 + 1 sinπ5
cos 2ϕ + 2 sin 2ϕ + 4 cos2 π 5 + tg π 5 + 3 sin2π5 , then π 5 Z 0 " 4 X i=1 Ai(ϕ) # dϕ = al 9 4− 3 cos 3 π 5 − 4 cos 2 π 5 − 9 2cos π 5 + 1 4 cos π5 − l2 4 9 2+ cos π 5 − 6 cos 2 π 5 + π 5 4 cos2π 5 + tg π 5 + 3 sin2π5 . (11) We obtain that: Pint = 10 πa2sinπ 5 2 cos π 5 + 1 2 al 9 4− 3 cos 3 π 5 − 4 cos 2 π 5 − 9 2cos π 5+ 1 4 cosπ5 −l 2 4 9 2 + cos π 5 − 6 cos 2 π 5 + π 5 4 cos2 π 5 + tg π 5 + 3 sin2π5 .
References
[1] D. Barilla, M. Bisaia, G. Caristi, A. Puglisi, On Laplace type problems (II), Far East Journal of Mathematical Sciences vol. 58 n. 2 , pp. 145-155. [2] D. Barilla, G. Caristi, A. Puglisi, A Laplace Type Problem for an Irreg-ular Trapetium Lattice and Body Test Rectangle, Applied Mathematical Sciences, Vol. 7, 2013, no. 10, 487 - 501.
[3] D. Barilla, G. Caristi, A. Puglisi, A Buffon - Laplace type problems for an irregular lattice and with maximum probabil-ity. Applied Mathematical Sciences, Vol. 8, no. 165, 8287-8293. http://dx.doi.org/10.12988/ams.2014.411916
[4] D. Barilla, G. Caristi, A. Puglisi, M. Stoka, Laplace type problems for a triangular lattice and different body test, Applied Mathematical Sciences, 2014, Applied Mathematical Sciences, Vol. 8, no. 103, pp. 5123 - 5131. http://dx.doi.org/10.12988/ams.2014.46423
[5] D. Barilla, G. Caristi, E. Saitta, A Laplace type problems for a lattice with cell composed by three quadrilaterals and with maximum proba-bility. Applied Mathematical Sciences, Vol. 8, no. 165, pp. 8279 - 8286. http://dx.doi.org/10.12988/ams.2014.411915
[6] D. Barilla, G. Caristi, E. Saitta, M. Stoka, Laplace type problem for lattice with cell composed by two quadrilaterals and one trian-gle, Applied Mathematical Sciences vol 8, 2014, no 16, pp. 789-804. http://dx.doi.org/10.12988/ams.2014.37407
[7] D. Barilla, A. Puglisi, E. Saitta, M. Stoka, A Laplace type problems for a lattice with cell composed by two triangles and a trapezium, Applied Mathematical Sciences, Vol. 8, 2014, no. 9, pp. 441 - 461. http://dx.doi.org/10.12988/ams.2014.37408
[8] D. Barilla, A. Puglisi, E. Saitta, A Buffon type problem for a lat-tice with fundamental cell composed by two triangles and two trapezi-ums. Applied Mathematical Sciences, Vol. 8, no. 165, pp. 8271 - 8278. http://dx.doi.org/10.12988/ams.2014.411914
[9] D. Barilla, A. Puglisi, E. Saitta, Laplace type problems for irregular lattice with cell ”Pentagon + Triangle”, International Mathematical Forum, Vol. 8, no. 23, pp. 1131 - 1142. http://dx.doi.org/10.12988/imf.2013.3232 [10] G. Caristi, Laplace Problem for Regular Lattices, Recent Advances in
Ap-plied Mathematics, editors Cornelia A. Bulacea, Valeri Mladenov, Monica Leba, Nikos Mastorakis, proceedings of the 14th WSEAS International Conference on Applied Mathematics Puerto de la Cruz, SPAIN, ISBN 9789604741380, pp. 27-33.
[11] G. Caristi, Buffon-Laplace Type Problems for Three Regular Lattices and ”Body Test” a Parallelogram, International Mathematical Forum, Vol. 8, 2013, no. 6, 245 - 271.
[12] G. Caristi, A. Puglisi, M. Stoka, A Laplace type problem for a regular lattices with octagonal cell, Far East Journal Mathematical Sciences vol. 48 n.1, pp. 103-118.
[13] G. Caristi, M. Stoka, A Laplace type problem for a regular lattices with irregular hexagonal cell, Far East Journal Mathematical Sciences vol. 50 n.1, pp. 23-36.
[14] G. Caristi, M. Stoka, A Buffon - Laplace type problem for an irregular lattice and ”body test” rectangle. Applied Mathematical Sciences, Vol. 8, no. 168, pp. 8395 - 8401. http://dx.doi.org/10.12988/ams.2014.411927 [15] H. Poincar´e, Calcul des probabilit´es, ed.2, Gauthier Villars, Paris, 1912. [16] M. Stoka , Probabilit´es g´eom´etriques de type Buffon dans le plan euclidien,
Atti Acc. Sci. torino, 110 (1975-1976), pp. 53-59.
