A Laplace type problem for a regular lattice with six obstacles

Transcription

1 Reent Advanes in Appied & Biomedia Informatis and Computationa Engineering in Systems Appiations A Lapae type probem for a reguar attie with six obstaes D. Baria, G. Caristi, A. Pugisi University of Messina Department S.E.A. Via dei Verdi, 75, 98- Messina Itay Abstrat: Let be R a, b, ; α) a reguar attie with the fundamenta e as in fig.. In this paper we ompute the probabiity that a random segment of onstant eght ersets a side of the attie. In partiuar when the fundamenta e beomes a retange we obtain the Lapae probabiity. Key Words: Geometri Probabiity, stohasti geometry, random sets, random onvex sets and egra geometry. AMS Cassifiation:6D5, 5A. Setion Let R a, b, ) be the reguar attie with fundamenta e is as in fig. : a O A A x B' y 9 A B'' A b B B 6 O 7 C 8 5 O 5 a b fig. b that a segment s erset a side of the fundamenta e C ) we have The obstaes are squares with the side of ength. Considering a random segment whose ength < min a, b) we to want ompute the probabiity that a segment ersets a side of the attie. This probabiity is equa to the probabiity P ) area C ) = ab 8 8 = ab. ) Considering now a position s of the segment with baryenter p and whih forms an ange with axis x. We onsider the imit positions of the segment s for a determinated ange, and et Ĉ) ) the figure determinated by these positions see fig. ): h F E E' F D' 6 D D'' 5 F 9 E F' O F E O fig. Let a, a,..., a 9 the poygons of the fig. suh that: areaĉ) ) = areac ) area a ) + area a ) + area a ) +area a ) + area a 5 ) + area a 6 ) + area a 7 ) +area a 8 ) + area a 9 ). ) Considering AB = x, A B = y, from triange AA B foow that x = tg, ) area a ) = x = tg. ) 8 From triange A A A we have that A A = tg, 5) O 6 ISBN:

2 Reent Advanes in Appied & Biomedia Informatis and Computationa Engineering in Systems Appiations E ' area a ) = A A = tg. 6) 8 F / Considering the 5), the paraeogram A B F F A F = a A A = a tg, / F h fig. E E / E h = os, area a ) = A F h = a ) os sin. 7) foow that E E = tg, F E = b E E = b tg, h = sin, area a 5 ) = F E h = b tg ) sin = In order to ompute the area of the poygon a onsidering the figure F b ) sin os. 9) In the same way we have that F ' / h C area a 8 ) = E E = tg. ) 8 F F From triange E D D we have that D D = tg, ) fig. We have F F F = =, h = sin + ) = sin + ) = area a 7 ) = D D = tg. ) and os + sin ), area a ) = h 8 = os + sin ) 8. 8) Compute now the area of the paraeogram a 5. From fig. From figure 5 D D D' h fig.5 D' ' ISBN:

3 Reent Advanes in Appied & Biomedia Informatis and Computationa Engineering in Systems Appiations we have D D = D D +, DD = E D =, and onsidering the ), we have that D D = tg +, and sine D DD =, From here B B = b x = b os, area a 9 ) = b ) os sin. 5) Considering the formue ), 6), 7), 8), 9), ), ), ) and ) in the ) we obtain that: h = D D sin ) = Then sin + os. area a 6 ) = tg + ) os = DD h = 8 sin + os ) 8. ) From fig. 6 A A x B ' ' foow that B ' a9 h fig.6 h = sin, B B / area a 9 ) = B B h = B B sin. From triange O B A we have that x + = os, ) x = os. areaĉ) ) = areac ) tg + tg + a ) os sin + os + sin ) tg+ b ) sin os + sin + os ) + tg + tg+ b ) os sin. 6) Considering A A = z, from the trianges AA B and A A A we obtain that y y = = sin, z = os, sin, z = os and sine y + z =, we have sin + os =. Mutipe the i two members of this reation before with sin and after with os, we obtain tg = sin, tg = os. Considering this reation in 5) we have that area Ĉ) ) = areac ) ISBN:

4 Reent Advanes in Appied & Biomedia Informatis and Computationa Engineering in Systems Appiations a + ) os + b ) sin sin. 7) Denoting with M the set of segments s whose baryenter are in C ) and N the set of segments s whose baryenter are in C ) that do not erset the boundary of C ), we have P ) = µ N ) µ M ), 8) where µ is the Lebesgue measure in Euidean pane. In order to ompute the measures µ M ) and µ N ) we use the Poinaré kinemati measure dk = dx dy, where x, y are the oordinates of p and the defined ange. Considering that,, we have that µ M ) = and by 6) µ N ) = x,y) C ) area C) 9) x,y) Ĉ) ) area Ĉ) ) = areac) a + ) + b ). ) Considering the 7), 8) and 9) we obtain the probabiity: = a + b ) ab = ) P ) a + b ) ab ) When the fundamenta e C ) beomes a retange with sides a and b and the obstaes beome pos and the probabiity P ) beomes the Lapae probabiity: Setion P = a + b + ). a b + ) Let be R a, b, ) the reguar attie with fundamenta e is as in fig. 7. Fig ) C fig.7 Denoting with C ) the fundamenta e, we have: area C ) = ab. ) The e C ) have six obstaes that are squares with diagona of eght with < min a, b). Considering a segment s of random position and of eght with < < min a, b), we want ompute the probabitiy that this segment ersets a side of attie. This probabiity is equa to probabiity P ) that the segment s ersets the bounderay of the fundamenta e. The position of the segment s is determinated by the midde po p and by the ange that the segment form with the axis x. We onsider the imit positions of the segment s that orrisponde at ange and et Ĉ) ) the determinated figure from this position see fig. 8): a b / / a a a a Fig ) C ) fig.8 b ISBN:

5 Reent Advanes in Appied & Biomedia Informatis and Computationa Engineering in Systems Appiations From this figure we an write: areaĉ) ) = areac ) areab ) + areab ) + areab ) + areab ). ) Considering some resuts that we have obtained in a previous paper, foow that: areaĉ) ) = ab a ) os + b ) sin + sin + ) os sin = ab a + ) os + b ) sin sin. ) Denoting with M the set of segments s whose the midde po are in C ) and N the set of segments s whose the midde po are in C ), we have that: P ) = µ N ) µ M ), ) where µ is the Lebesgue measure in Euidean pane. In order to ompute the measures µ M ) and µ N ) we use the Poinaré kinemati measure dk = dx dy, area C ) ) = ab ) a + ) os + b ) sin sin = ab ) a + b ). 6) Considering the ), ) and 5) we obtain the probabiity: P ) a + b) = ab. 7) ) When, the obstaes beames pos and the fundamenta e beames a retange with side a and b. In this ase the probabiity 7) beomes the Lapae probabiity: Setion P = a + b). ab Let R a, b, ; α) be the reguar attie with fundamenta e C ) is as in fig.. a α b ) C ) b where x, y are the oordinates of p and the defined ange. Sine,, we obtain that: Fig 7 fig.9 µ M ) = x,y) C ) areac) = ab ). 5) and onsidering the ) µ N ) = x,y) Ĉ) ) where α, is an ange and < min a, b). The C ) have six obstaes that are rhombs with side and with the diagonas d = sin α, d = os α. We have that: areac ) = ab + a os α + b sin α) + sin α os α. In the same way of the Setion, denoting with Ĉ ) ) the figure determinated by the imit positions of the segments s for a assigned vaue of see fig. ): ISBN:

6 Reent Advanes in Appied & Biomedia Informatis and Computationa Engineering in Systems Appiations m m m m m5 ) C ) sin tgα os tgα) = areac) a + b + os α + sin α) 5 fig. Fig 8 From this figure we an write: sin α) + areaĉ) ) = areac ) a sin α) os α os + b sin + sin + α) + sin α) = areac ) a + sin α + os α) os + b+ os α + sin α) sin sin tgα os tgα) 8) Denoting with M the set of segments s whose the midde po are in C ) and N the set of segments s whose the midde po are in C ), we have that: P ) = µ N ) µ M ). 9) Sine,, we obtain that: tgα). ) Considering the 9) and ) in 8) we obtain that: P ) = a + b + os α + sin α) ab + a os α + b sin α) + sin α os α. ) When, the obstaes beames segments of enght that go in the ounderay of the attie, the fundamenta e C ) beames a retange of side a and b + and the ) beomes the Lapae probabiity: Referenes: P = a + b + ). a b + ) Caristi G and Stoka M., A Lapae type probem for a reguar attie with obstaes I), Atti A. Si. Torino, to appear). Caristi G and Stoka M., A Lapae type probem for a reguar attie with obstaes II), Atti A. Si. Torino, to appear). Poinaré H., Cau des probabiitiés, ed., Carré, Paris, 9. Stoka M., Probabiités géométriques de type Buffon dans e pan euidean, Atti A. Si. Torino, T., pp. 5-59, µ M ) = x,y) C ) and areac), ) µ N ) = areac) x,y) Ĉ) ) a + sin α + os α) os + b + os α + sin α) sin ISBN: