A Laplace Type Problems for a Lattice with Cell Composed by Two Triangles and a Trapezium

In some previous papers [1], [2], [3] and [4] the authors studies same Laplace problems with different fundamental cells. In this paper we consider a lattice with fundamental cell rapresented in fig.1 and we compute the probability that a segment of random position and constant length intersects a side of lattice.


Introduction and Main Results
Let (a; m) where 0 < m < 2a cos π 5 the lattice with the fundamental cell C 0 rappresented in fig. 1 A (1) We want to compute the probability that a segment s with random position and of constant length l < a 4 cos π 5 intersects a side of lattice , i.e. the probability P int that the segment s intersects a side of fundamental cell C 0 .
The position of the segment s is determinated by the center and by the angle ϕ that it forms with line CE.
To compute the probability P int we consider the limiting positions of segment s, for a specified value of ϕ.
So, we have the fig. 2 fig. 2 and the formulas At first we consider the fig. 3 fig. 3 We have With these angles, from the trinagle BB 4 B 5 we have then and, considering the (3), we have give us Therefore, by second formula (8) follows Therefore, we see that The relations (9) and (10) give us Now we consider the fig. 5 Hence we can write By fig.6 then The fig. 7 7 give us and, considering the formulas ( 1), ( 8) and ( 13), By the fig.8 fig. 8 we have Moreover, considering the second formula (13) we have Consequently By relations ( 14) and ( 16) we have Denoting with and, combining the formulas (11), ( 12), ( 15) and ( 17), we obtain With this valor, the relation (5) become Now we compute area C 02 (ϕ).The fig.9 Then Now we consider the fig. 10 We have Then, considering the formula (2) we have By the fig. 11 fig. 11 we have Then, considering the relation (20), we have Therefore The formulas (21) and (23) give us Now we consider the fig. 12 fig. 12 we have The fig. 13 give us With these angles, from triangle DD 1 D 7 we have fig. 14 We have and, considering the formulas (26) and ( 27), we can write Therefore, we see that The fig.15 fig. 15 and the first formula (20) give us then At last, we consider the fig. 16 fig. 16 Hence we have that Denoting with the formula ( 6) can be write Combining the relations ( 22), ( 24), ( 25), ( 28), ( 29), ( 30) and (31) we obtain we have Hence we have and, with the second formula (35), Then The relations (36) and (37) give us By the fig.20 we have Therefore Now we consider the fig.21 We have Considering the formula (39), we can write This relation with the (40) give us At last, we consider the fig. 22 We have Therefore, we see that Considering the relations (34), (38), ( 41) and (42) we can write Denoting with M i (i = 1, 2, 3) the set of all segments s which have center in cell C 0i denote likewise with N i the set of all segments s completely contained in C 0i , we have [2]: where μ is the Lebesgue measure in the euclidean plane.
To compute the measure μ (M i ) and μ (N i ) we use the kinematic measure of Poincaré [5] dk = dx ∧ dy ∧ dϕ, where x, y are the coordinate of center of s and ϕ the specified angle.