Geometric Probabilities for an Arbitrary Convex Body of Revolution in E 3 and Certain Lattice

Transcription

1 Int. J. Contemp. Mth. Sciences, Vol. 4, 29, no. 25, Geometric Probbilities for n Arbitrry Convex Body of Revolution in E nd Certin Lttice Giuseppe Cristi Diprtimento di discipline economico ziendli Fcoltà di Economi, Università di Messin Vi dei Verdi 75, Me Itly gcristi@dipmt.unime.it Abstrct In this note we solve problems of Buffon type for n rbitrry convex body of resolution K in the eucliden spce E nd prticulr lttice R. As prticulr cse we study the probbility of intersection between rndom sphere nd the sides of R. Mthemtics Subject Clssifiction: 6D5, 52A22 Keywords; Geometric Probbility, stochstic geometry, rndom sets, rndom convex sets nd integrl geometry Problems of geometric probbility for n rbitrry convex body of resolution in the eucliden spce E hs been investigted in [1]. In [9] Buffon s problem is solved for lttice of right-ngled prllelepipeds in the -dimensionl spce. In this note we wnt to use the results in [5] for to solve problems of intersection for prticulr lttice tht we describe: the fundmentl cell C of the lttice R is right-ngled prism of height c nd whose bsis is the following: Let K be n rbitrry convex body of resolution with centroid G nd oriented xis of rottion r. The line r is determined by the ngle ϑ between r nd the z-xis nd by the ngle ϕ between the projection of r on the xy-plne nd the x-xis. Hence r = rϑ, ϕ. Then the length L of the projection of K on the z-xis is given by Lϑ, ϕ =pϑ, ϕ+pπ ϑ, ϕ

2 124 G. Cristi B E F T A L M C N H G D C,π : Bsis of the prism of the fundmentl cell C. where pϑ, ϕ is the distnce from G to the xy-plne when K is tngent to the xy-plne. Now let C be fundmentl cell of the lttice R nd ssume tht the two -dimensionl rndom vribles defined by the coordintes of G nd by the tringle ϑ, ϕ, ψ re uniformly distributed in the cell C nd in [,π] [, 2π] [, 2π] respectively. We denote by M C the set of ll test bodies K whose centroid G lies in C nd by N C the set of bodies K tht re completely contined in C. We wnt to compute the probbility p K,R tht the body K intersects the lttice R. Denoting with μ the Lebesgue mesure, the probbility is given by p K,R =1 μn C μm C. 1 Consider for fixed ϑ, ϕ [,π] [, 2π] the set of points P C for with the body K with centroid P nd rottion xis r does not intersect the boundry C nd let Cϑ, ϕ the topologicl closure of this open subset of C. We will ssume tht the body K is smll 1 with respect to the lttice R. Denoting with DimK the dimeter of the body K, using the generl result in [5], K, is sid smll respect to R iff DimK < Using the kinemtic mesure see:[9] c, 8 +. dk = dx dy dz dω dψ, 2 1 We sy tht the body K is smll with respect to R, if the polyhedrons sides of Cϑ, ϕ nd C re pirwise prllel.

3 Geometric probbilities for n rbitrry convex body 125 where x, y, z re the coordintes of G, dω = sin ϑdϑ dϕ, nd ψ is ngle of rottion bout r. IfS E is mesurble subset we denote with vols the Eucliden volume of S. We hve see:[5] μm C =8π 2 volc, μn C =2π 2π π volcϑ, ϕ sin ϑdϑ dϕ. 4 Hence p K,R =1 1 4πvolC π volcϑ, ϕ sin ϑdϑ dϕ. 5 Theorem 1. If K is smll with respect to R, the probbility p K,R is given by where: p K,R =1 Λ 8 π 2 c. 6 nd 1 Lϑ, ϕ c Λ:= π Lϑ 1 ϑ, ϕ,ϕ 1 ϑ, ϕ Lϑ 2 ϑ, ϕ,ϕ 2 ϑ, ϕ c Lϑ, ϕ+ 1 2p 1 2p 2 4p 2 2 volc sin ϑdϑ dϕ. cot ϑ ϑ 1 ϑ, ϕ := rccossin ϑ cos ϕ, ϕ 1 ϑ, ϕ := rctn, sin ϕ ϑ 2 ϑ, ϕ := rccos sin ϑ sin ϕ, ϕ 2 ϑ, ϕ := rctn tn ϑ sinϕ + π/2, ϑ ϑ, ϕ := rccos sin ϑ sinϕ + π/6, with ϕ ϑ, ϕ := tnϑ cosϕ 2 + π/6, p 1 := pϑ 1 ϑ, ϕ,ϕ 1 ϑ, ϕ, p 2 := pϑ 2 ϑ, ϕ,ϕ 2 ϑ, ϕ, p := pϑ ϑ, ϕ,ϕ ϑ, ϕ.

4 126 G. Cristi Proof: We use the following nottions: N 1 the set of ll test bodies of revolution K completely contined in the prism whose bse is the rectngle EHGF we sy this prism C 1 ; N 2 the set of ll test bodies of revolution K completely contined in the prism tringle whose bse is the tringle ALH we sy this prism C 2 ; Consider for fixed ϑ, ϕ [,π] [, 2π] the set of points P C for with the body K with centroid P nd rottion xis r does not intersect the boundry C i i =1, 2 nd let C i ϑ, ϕ i =1, 2 the topologicl closures of this open subsets of C i. Then formul 1 becomes p K,R =1 μn 1+8μN 2, 7 μm C Now, let us consider the cell C 1 with the coordintes ϑ 1,ϕ 1,ψ 1. Follows [5] we put: ϑ 1 1 ϑ 1,ϕ 1 := rccossin ϑ 1 cos ϕ 1, ϕ 1 ϑ1 1 ϑ 1,ϕ 1 := rctn, sin ϕ 1 ϑ 1 2 ϑ 1,ϕ 1 := rccos sin ϑ 1 sin ϕ 1, ϕ 1 2 ϑ 1,ϕ 1 :=rctn tn ϑ 1 sinϕ 1 + π/2, Hence we obtin the expression of the volume: volc 1 ϑ 1,ϕ 1 = Lϑ 1 1 ϑ 1,ϕ 1,ϕ 1 1 ϑ 1,ϕ 1 Lϑ 1 2 ϑ 1,ϕ 1,ϕ 1 2 ϑ 1,ϕ 1 c Lϑ 1,ϕ 1, Let us consider the cell C 2 with the coordintes ϑ 2,ϕ 2,ψ 2. We denote: ϑ 2 1 ϑ 2,ϕ 2 := rccossin ϑ 2 cos ϕ 2, ϕ 2 ϑ2 1 ϑ 2,ϕ 2 := rctn, sin ϕ 2 ϑ 2 2 ϑ 2,ϕ 2 := rccos sin ϑ 2 sin ϕ 2,

5 Geometric probbilities for n rbitrry convex body 127 ϕ 2 2 ϑ 2,ϕ 2 := rctn tn ϑ 2 sinϕ 2 + π/2, ϑ 2 ϑ 2,ϕ 2 := rccos sin ϑ 2 sinϕ 2 + π/6, ϕ 2 ϑ 2,ϕ 2 := tn ϑ 2 cosϕ 2 + π/6. Hence we give the volume of C 2 : volc 2 ϑ 2,ϕ 2 = 1 Lϑ 2,ϕ 2 c 1 2p 1 2p 2 4p 2 2 volc. After chnging of vribles ϑ i,ϕ i,ψ i ϑ, ϕ, ψ, for i =1, 2 we compute: p K,R =1 μn 1+8μN 2 = 8 μm C 1 [ 2π π =1 volc 1 ϑ 1,ϕ 1 sin ϑ 1 dϑ 1 dϕ 1 + 4πvolC π +8 volc 2 ϑ 2,ϕ 2 sin ϑ 2 dϑ 2 dϕ 2 ]. With the following position: Λ:= π Lϑ 1 ϑ, ϕ,ϕ 1 ϑ, ϕ Lϑ 2 ϑ, ϕ,ϕ 2 ϑ, ϕ c Lϑ, ϕ+ 1 Lϑ 2,ϕ 2 1 2p 1 c 2p 2 4p 2 2 volc sin ϑdϑ dϕ. We obtin p K,R =1 Λ 4πvolC. 9 As ppliction of the theorem we cn compute the probbility of intersection with side of the lttice R when K is rndom sphere Σ of constnt rdius R nd D s dimeter.

6 128 G. Cristi Corollry 2. If Σ is smll with respect to R, the probbility p Σ,R is given by 1 { p Σ,R =1 2 D Dc D c [ 8 + ][ R ] } R c R. Remrk. It is possible to obtin the result in Corollry 2 using theorems of E.Bosetto in []. References [1] A. Alemn. M.I. Stok,M. Zmfirescu. Convex bodies insted of nedles in Buffon s experiment, Geometrie Dedict , 1-8. [2] A. Arc. A. Dum, Two Buffon-Lplce type problems in the Eucliden plne, Suppl. Rend. Circ. Mt. Plermo II, nr , pp [] E. Bosetto, Geometric Probbolities in the Eucliden Spce E, Disserttion, Fernuniversität Hgen [4] A. Dum. M.I. Stok, Geometric Probbilities for Convex Bodies of Lrge Revolution in the Eucliden Spce E, Suppl. Rend. Circ. Mt. Plermo II, nr.65 2, pp [5] A. Dum. M.I. Stok, Geometric Probbilities for Convex Bodies of Lrge Revolution in the Eucliden Spce E II, Beiträge zur Algebr und Geometrie Contributions to Algebr nd Geometry. Vol. 4 22, nr. 2, [6] H. Minkowski., Volumen und Oberfläche, Mth. Ann., 59, 19. [7] H. Poincré., Clculus des probbilities, ed.2, Crré, Pris, [8] H. Solomon, Geometric Probbility, Soc. for Ind. nd Appl. Mth., [9] M.I. Stok, Un extension du problème de l iguille de Buffon dns l espce euclidien R n, Boll. Unione Mt. Itlin , Received: June, 28