Dedicted to Professor Octv Onicescu, founder of the Buchrest School of Probbility LAPLACE TYPE PROBLEMS FOR A DELONE LATTICE AND NON-UNIFORM DISTRIBUTIONS G CARISTI nd M STOKA Communicted by Mrius Iosifescu This work considers Delone lttice with the fundmentl cell represented in gure 1 The probbility is determined tht constnt-length segment with rndom exponentil distribution direction nd γ () will intersect side of the lttice AMS 1 Subject Clssiction: 6D5, 5A Key words: geometric probbility, stochstic geometry, rndom sets, rndom convex sets nd integrl geometry 1 PRELIMINARIES Let R (, α) with π α π 3 be Delone lttice with the fundmentl cell C represented in Fig 1 This reltion leds to the following other reltions BAD = π α, DGE = DGF = π α, (1) BAC = π α, ÊGF = α BD = CD = cos α, EG = F G = ctgα, () AG =, DG = (3) re C = We wnt to determine the probbility tht segment s with constnt length l with l < 3 6 nd with rndom non-uniform distribution length, will intersect side of the lttice R, ie the probbility P int tht the segment s will intersect side of the fundmentl cell C REV ROUMAINE MATH PURES APPL 6 (17),, 51958
5 G Cristi nd M Stok A C 1 F π π E G C C 3 B α D α C Fig 1 MAIN RESULTS Let ϕ be the ngle between the segment s nd the line BC By considering the limit positions of the segment s in the cell C for given ϕ, we obtin Fig A 1 A A 1 A 3 C ɵ 1(ϕ) 6 F b 1 F 3 F F 1 F b 6 G c 1 G 1 E 5 E 3 E 1 3 5 G E 6 E E E c 6 b C ɵ (ϕ) b 5 c C ɵ 3(ϕ) c 5 B B 1 b b ϕ 3 ϕ ϕ D 1 D D D 3 D C 1 C 3 c 3 ϕ c C C Fig
3 Lplce type problems for Delone lttice nd non-uniform distributions 51 And the formuls () re Ĉ1 (ϕ) = re C 1 (5) re Ĉ (ϕ) = re C (6) re Ĉ3 (ϕ) = re C 3 Figure nd formul (1) led to (7) A 1 AA = π α, re i (ϕ), re b i (ϕ), re c i (ϕ) AA 1 A = F 3 BB 1 = α ϕ, With these ngles, the tringle AA 1 A leds to (8) AA 1 = l sin (ϕ + α), AA = (9) re 1 (ϕ) = l sin (α ϕ) sin (ϕ + α) Moreover, from Fig we obtin (1) ϕ [, α] Figure nd reltions (7) nd (8) led to h = l sin (α ϕ), A 1F = AA 1 = l sin (α ϕ), AA A 1 = ϕ + α l sin (ϕ + α), (11) re (ϕ) = l sin (α ϕ) l sin (α ϕ) sin (ϕ + α) From Fig nd formuls () nd (7), we obtin F 1 F G = π AA 1 A = π (α ϕ), h 3 = l cos (α ϕ) (1) re 3 (ϕ) = F G h 3 = l ctgα cos (α ϕ) Similrly, gure nd reltion (7) led to (13) EE E 1 = π AA A 1 = π (ϕ + α), EE 1 E = α + ϕ π,
5 G Cristi nd M Stok (1) EE 1 = l sin (ϕ + α), EE = l cos (ϕ + α) (15) re 5 (ϕ) = l sin (ϕ + α) cos (ϕ + α) Lemm 1 Tking into ccount reltion (1) nd (1) we obtin cos (ϕ + α) Figure nd formuls (), (13) nd (1) led to E GG = EE 1 E = ϕ+α π, h = l ( ) sin E GG = l cos (ϕ + α), GE 1 = EG EE 1 = ctgα l sin (ϕ + α) (16) re (ϕ) = l ctgα cos (ϕ + α) + l sin (ϕ + α) Finlly, Fig nd formuls (7), (8) nd (1) led to A 3 A E = π AA A 1 = π (ϕ + α), h 6 = l sin (ϕ + α), A E = AA EE = l sin (α ϕ) (17) re 6 (ϕ) = l [ sin (ϕ + α) l sin (α ϕ) sin (ϕ + α) + l cos (ϕ + α), From formuls (9), (11), (1), (15), (16) nd (17) we obtin (18) A 1 (ϕ) = ] cos (ϕ + α) re i (ϕ) = l sin (ϕ + α) + l ctgα [1 cos (ϕ + α)] By substituting this expression into (), it follows tht (19) re Ĉ(ϕ) = re C 1 A 1 (ϕ) Figure leds to () ÊF 3 F = α ϕ, F F F 3 = π (α ϕ) (1) F F = l sin (α ϕ), F F 3 = l cos (α ϕ)
5 Lplce type problems for Delone lttice nd non-uniform distributions 53 () re b 1 (ϕ) = l sin (α ϕ) Figure nd formuls () e (1) led to h = l sin (α ϕ), BF 3 = F F 3 = l cos (α ϕ), (3) re b (ϕ) = l sin (α ϕ) l sin (α ϕ) Similrly, we obtin () DD 1 = l cos ϕ, DD = l sin ϕ, (5) re b (ϕ) = l sin ϕ From Fig nd reltions () nd (), it follows tht h 3 = l sin(ϕ), BD 1 = BD DD 1 = cos α l cos ϕ (6) re b 3 (ϕ) = l cos α sin ϕ l sin ϕ Figure nd formuls () e () led to h 5 = l cos ϕ, D G = DG DD = l sin ϕ, tht is (7) re b 5 (ϕ) = l cos ϕ l cos ϕ l sin ϕ Finlly, from Fig nd reltions () nd (1), it follows tht h 6 = l cos (α ϕ), F G = F G F F = ctgα l sin (α ϕ), (8) re b 6 (ϕ) = l ctgα cos (α ϕ) l sin (α ϕ) Tking into ccount formuls (), (3), (5)(7) nd (8), we obtin (9) A (ϕ) = re b i (ϕ) = l sin (ϕ + α) l [ cos ϕ + (1 cos α) sin ϕ]
5 G Cristi nd M Stok 6 By substituting this expression into (5), it follows tht (3) re Ĉ (ϕ) = re C 1 A (ϕ) From Fig we obtin (31) E 1 GE 5 = π α, E 1 E 5 G = π ϕ, With these ngles, from tringle E 1 E 5 G it follows tht (3) E 1 G = l cos ϕ, E 5G = E 5 E 1 G = ϕ + α π l cos (ϕ + α) (33) re c 1 (ϕ) = l cos ϕ cos (ϕ + α) Tking into ccount Fig nd formuls () nd (3), we obtin h = l cos ϕ, DE 5 = DG E 5 G = l cos (ϕ + α) +, (3) re c (ϕ) = l cos ϕ l cos ϕ + l cos ϕ cos (ϕ + α) Figure leds to (35) CC 1 = l sin ϕ, CC = l sin (ϕ + α) (36) re c (ϕ) = l sin ϕ sin (ϕ + α) From Fig nd reltions () nd (35), it follows tht h 3 = l sin ϕ, C D = CD CC = cos α (37) re c 3 (ϕ) = l cos α sin ϕ l sin ϕ sin (ϕ + α) Figure nd formul (35) led to l sin (ϕ + α), E 6 C 3 C 1 =π (ϕ + α), h 5 = l sin (ϕ + α), C 1E = CC 1 = l sin ϕ, tht is (38) re c 5 = l sin (ϕ + α) l sin ϕ sin (ϕ + α)
7 Lplce type problems for Delone lttice nd non-uniform distributions 55 Finlly, tking into ccount Fig nd reltions () nd (3), we obtin (39) re c 6 = lctgα cos (ϕ + α) + l cos ϕ cos (ϕ + α) From formuls (33), (3), (36)(38) nd (39), it follows tht () A 3 (ϕ) = re c i (ϕ) = l (cos α sin ϕ + cos ϕ) l cos ϕ l (sin ϕ ctgα cos ϕ) By substituting this expression into (6), we obtin (1) re Ĉ3 (ϕ) = re C 3 A 3 (ϕ) Let M i (i = 1,, 3) be the set of segments s the midpoint of which is in cell C i nd let N i be the set of segments s tht re entirely contined in cell C i We obtin [3]: () P int = 1 3 µ(n i ), 3 µ(m i ) where µ is the Lebesgue mesure of the Eucliden plne Mesures µ(m i ) nd µ(n i ) re determined by using the Poincr e kinemtic mesure []: dk = dx dy dϕ, where x, y re the coordintes of the midpoint of s nd ϕ is the ngle dened bove Let us ssume tht the direction of the support line of s is rndom vrible with probbility density of f (ϕ) Tking into ccount formul (1), we obtin (3) µ (M i ) = f (ϕ) dϕ dxdy {(κ,y) C i } = (re C i ) f (ϕ) dϕ = (re C i ) f (ϕ) dϕ nd tking into ccount formuls (19), (3) nd (9), we obtin π [ ] µ (N i ) = f (ϕ) dϕ {(κ,y) Ĉi(ϕ)} dxdy = re Ĉi(ϕ) f (ϕ)
56 G Cristi nd M Stok 8 = [re C i A i (ϕ)] f (ϕ) dϕ = (re C i ) These formuls led to 3 µ (M i ) = (re C ) 3 µ (N i ) = (re C ) f (ϕ) dϕ f (ϕ) dϕ f (ϕ) dϕ, [ 3 ] A i (ϕ) f (ϕ) dϕ By substituting these expressions into (), we obtin 1 α [ 3 ] () P int = A α i (ϕ) f (ϕ) dϕ (re C ) f (ϕ) dϕ From formuls (18), (9) nd (), it follows tht 3 A i (ϕ) = l ( cos ϕ + cos α sin ϕ) l cos ϕ l A i (ϕ) f (ϕ) dϕ [( + ctgα ctgα) cos ϕ + (3 cos α) sin ϕ ctgα (1 cos α)] With this vlue (3), reltion () cn be written s (5) P int = 1 α ( ) f (ϕ) dϕ { ( cos ϕ + cos α sin ϕ) cos ϕ l [( + ctgα ctgα) cos ϕ + (3 cos α) sin ϕ ctgα (1 cos α)]} f (ϕ) dϕ 1 Exponentil distribution We hve f(ϕ) = e ϕ In previous pper [1], we demonstrted the following formuls: f (ϕ) dϕ = 1 e ϕ,
9 Lplce type problems for Delone lttice nd non-uniform distributions 57 f (ϕ) sin ϕdϕ = 1 1 ( + cos α) e α, f (ϕ) cos ϕdϕ = 1 + 1 ( cos α) e α, f (ϕ) sin ϕdϕ = 1 5 f (ϕ) cos ϕdϕ = 1 5 [ ( + cos α) e α ], [ 1 + ( cos α) e α ] By substituting these vlues into reltion (5), we obtin l ( ) (1 e α ) P int = { [ + cos α ( + cos α) e α] [ 1 + ( cos α) e α ] l [6 + cos α ctgα ctgα + 5ctgα cos α + e α ( + cos α + sin α 3 cos α +5ctgα + ctgα cos α 6ctgα cos α)]} Demonstrtion γ We hve f(ϕ) = ϕe ϕ In previous pper [1], we demonstrted the following formuls: f (ϕ) dϕ = 1 (1 + α) e α, f (ϕ) sin ϕdϕ = 1 1 e α cos α α e α ( + cos α), f (ϕ) cos ϕdϕ = 1 e α ( + cos α) + α e α ( cos α), f (ϕ) sin ϕdϕ = 5 + 1 5 e α (3 + 8 cos α) + αe α ( cos α),
58 G Cristi nd M Stok 1 f (ϕ) cos ϕdϕ = 1 5 + 1 5 e α ( + 9 cos α) + αe α ( cos α) By substituting these vlues into reltion (5), we obtin l P int = [1 (1 + α) e α ] ( [ e α ( cos α 1 ctgα 1 ) αe α ( + cos α 1 ctgα + 1 ] ) + cos α l { 1 5 + 1 5 8 5 cos α 5 ctgα [ 9 5 ctgα + cos αctgα + 8 3 e α + cos α + cos α 5 5 5 16 5 cos α 16 5 cos α + 5 + 9 cos αctgα 5 18 ] cos αctgα + (1 + α) (1 cos α) ctgα + αe α ( sin α 5 3 cos α + 3 5 cos α cos αctgα + cos αctgα)}) REFERENCES [1] G Cristi nd M Stok, A Lplce type problems for tringulr lttice nd non-uniform distributions Fr Est J Mth Sci 99 (16), 1383139 [] H Poincr e, Clcul des probbilit es, ed, Guthier Villrs, Pris, 191 [3] M Stok, Probbilit es g eom etriques de type Buon dns le pln euclidien Atti Acc Sci Torino 11 (19751976), 5359 Received 8 September 15 University of Messin, Deprtment of Economics, Vi dei Verdi, 75 981, Messin gcristi@unimeit Science Acdemy of Turin Vi Mri Vittori, 3 113 Torino, Itly mriusstok@gmilcom