PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY GEOMETRIC PROBABILITY CALCULATION FOR A TRIANGLE

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1 PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physica and Mathematica Sciences 07, 5(3, p. 6 M a t h e m a t i c s GEOMETRIC PROBABILITY CALCULATION FOR A TRIANGLE N. G. AHARONYAN, H. O. HARUTYUNYAN Chair of the Theory of Probabiity and Mathematica Statistics YSU, Armenia Let P(L(ω D is the probabiity that a random segment of ength in R n having a common point with body D entirey ies in D. In the paper, using a reationship between P(L(ω D and covariogram of D the expicit form of P(L(ω D for arbitrary triange on the pane is obtained. MSC00: Primary 60D05; Secondary 5A, 53C65. Keywords: covariogram, kinematic measure, orientation-dependent chord ength distribution, convex body, triange. Introduction. Let R n (n be the n-dimensiona Eucidean space, D R n be a bounded convex body with inner points, and V n be the n-dimensiona Lebesgue measure in R n. Consider the set of the segments of a constant ength that are contained in D. The measure evauation probem of such segment sets no simpe soution and depends on the shape of D. It is known the expicit form for the kinematic measures of the disk, the rectange, if the ength of the segment is ess than the smaer side of the rectange (see [, ], the equiatera triange, the rectange and the reguar pentagon (for an arbitrary ength of the segment [3]. D e f i n i t i o n. (see []. The function C(D,h = V n (D (D + h, h R n, is caed the covariogram of the body D. Here D + h = {x + h, x D}. Let S n denote the (n -dimensiona unit sphere in R n centered at the origin. We consider a random ine, which is parae to u S n and intersects D, that is, an eement from the set: Ω (u = {ines, which are parae to u and intersect D}. Let Πr u D be the orthogona projection of D onto the hyperpane u (here u stands for the hyperpane with norma u and passing through the origin. E-mai: narine78@ysu.am E-mai: harutyunyan.hripsime@ysu.am

2 Proc. of the Yerevan State Univ., Phys. and Math. Sci., 07, 5(3, p. 6. A random ine, which is parae to u and intersects D, has an intersection point (denoted by x with Πr u D. We can identify the points of Πr u D and the ines, which intersect D and are parae to u, meaning that we can identify the sets Ω (u and Πr u D. Assuming that the intersection point x is uniformy distributed over the convex body Πr u D, we can define the foowing distribution function. D e f i n i t i o n. The function F(u,t = V n {x Πr u D : V (g(u,x D < t} b D (u is caed orientation-dependent chord ength distribution function of D in direction u at a point t R, where g(u,x is the ine, which is parae to u and intersects Πr u D at the point x and b D (u = V n (Πr u D. Observe that each vector h R n can be represented in the form h = (u,t, where u is the direction of h, and t is the ength of h. Let L(ω be a random segment of ength > 0, which is parae to a given fixed direction u S n and intersects D. Consider the random variabe L (ω := V (L(ω D, where L(ω Ω (u, and the set Ω (u is defined as foows: Ω (u = {segments of engths, which are parae to u and intersect D}. Observe that each random segment L(ω ying on a ine g(u,x can be specified by the coordinates (g(u, x, y, where y is the one-dimensiona coordinate of the center of L(ω on the ine g(u,x. As the origin on the ine g(u,x we take one of the intersection points of the ine g(u, x with the boundary of domain D. Using the above notation, we can identify Ω (u with the foowing set: Ω (u = { (x,y : x Πr u D, y [, χ(u,x + ]}, where χ(u,x = V (g(u,x D. Note that the set Ω (u does not depend on the choice of the origin of the ine g(u,x, and the choice of the positive direction foows from the expicit form of the range of y. Further, we set B u,t D = { (x,y Ω (u : L (x,y < t }, t R, and observe that the sets Ω (u and B u,t D are measurabe subsets of Rn. D e f i n i t i o n 3. The function F L (u,t = V n(b u,t D V n (Ω (u = dxdy V n (Ω (u B u,t D is caed orientation-dependent distribution function of the ength of a random segment L in direction u S n. Let G n be the space of a ines g in R n. A ine g G n can be specified by its direction u S n and its intersection point x in the hyperpane u. The density du is the voume eement du of the unit sphere S n, and dx is the voume eement on u at x. Let µ( be a ocay finite measure on G n, invariant under the group of Eucidian motions. It is we known that the eement of µ( up to a constant factor has the foowing form (see []: µ(dg = dg = dudx.

3 Aharonyan N.G., Harutyunyan H.O. Geometric Probabiity Cacuation for a Triange. 3 Denote by O n = σ n (S n the surface area of the unit sphere in R n. For each bounded convex body D, we denote the set of ines that intersect D by We have (see [] [D] = {g G n, g D /0}. µ([d] = O n V n ( D. (n A random ine in [D] is the one with distribution proportiona to the restriction of µ to [D]. Therefore, for any t R we have F(t = µ({g [D], V (g D < t}, µ([d] which is caed the chord ength distribution function of D. Let L be a random segment of ength in R n and et K( be the kinematic measure of L []. If g G n is the ine containing L and y is the one-dimensiona coordinate of the center of L on the ine g, then the eement of the kinematic measure up to a constant factor is given by dk = dgdydk [], where dy is the one-dimensiona Lebesgue measure on g and dk [] is a motion eement in R n that eaves g unchanged (see [, 4 7]. Note that in the case, where the segment is orientated, the constant factor is equa to, whie for the unoriented segment it is equa to /. In this paper we consider ony the case of unoriented segments. The ength L of a random segment L, provided that it hits the body D, has the foowing distribution function: F L (t = K(L : L D /0,V (L D < t, t R. K(L : L D /0 Denote by P(L(ω D probabiity, that random segment of ength in R n having a common point with body D entirey ying in body D (in this case the direction of the segment L(ω is arbitrary. P r o p o s i t i o n (see [7]. Probabiity P(L(ω D in terms of chord ength distribution function F(t has the foowing form: ( O n V n ( D F(zdz + (n O n V n (D 0 P(L(ω D =. (n O n V n (D + O n V n ( D Case of a Triange. For any body D of the R n we have (see [7] P(L(ω D = C(D,u, O n S n V n (D + b D (u du, whie the kinematic measure of the segments entire ying in D is cacuated by the foowing formua: K(L(ω D = C(D,u,du. S n For any panar bounded convex domain we have P(L(ω D = π C(D,u,du. ( πs(d + D 0

4 4 Proc. of the Yerevan State Univ., Phys. and Math. Sci., 07, 5(3, p. 6. Denote by a triange in the pane. The main resut of the present paper is the foowing statement. T h e o r e m. Probabiity P(L(ω for arbitrary triange has the expicit forms ( (8 depending on the vaue of. P r o o f. Without oss of the generaity we assume, that AB a is the ongest side of ABC, CAB α is the smaest ange, and ABC β. Thus, we have BC = sin(α + β, CA = asinβ, BCA = π (α + β. Since AB is the argest sin(α + β side, then BCA is the biggest ange. Therefore α β π (α + β. Covariogram of a triange with side a has the form (see [3]: (asinβ t sin(u + β sinα asinβ, u [0,α],t [0, sinβ sin(α + β sin(u + β ], ( t sinu sin(α + β, u [α,π β], sinα sinβ sin(α + β t [0, sin(α + βsinu ], ( t sin(u α sinβ, u [π β,π], t [0, sinα sin(α + β sin(u α ], C(,u, = (asinβ +t sin(u + β sinα, u [π,π + α], t [0, asinβ sinβ sin(α + β sin(u + β ], ( +t sinu sin(α + β, u [π + α,π β], sinα sinβ sin(α + β ( +t sin(u α sinβ sinα sin(α + β Let consider the foowing cases a 0 sin(α + β. Using (, we get P(L(ω = t [0, sin(α + βsinu ],, u [π β,π], t [0, sin(u α ]. π C(,u,du = πs( + 0 sin(α + β πa sinα sinβ + a(sinα + sinβ + sin(α + β + π β α ( α ( sinu sin(α + β π du+ sinα sinβ sin(α + β π β 0 (asinβ sin(u + β sinα du+ sinβ sin(α + β ( sin(u α sinβ sinα sin(α + β du.

5 Aharonyan N.G., Harutyunyan H.O. Geometric Probabiity Cacuation for a Triange. 5 We set f (x,y sinα y sinβ ( y + x 4a sinα sin + β sin f (x,y 4a sin(α +βsin x (asinβ sin(u + β du = a sinα sinβ(y x ( y x + sinα sinβ ((y x sin(y xcos(y+x+β, y ( sinu sin(α + β du = a sinα sinβ(y x sinα sinβ x ( ( y + x y x sin + sin (α + β sinα sinβ ((y x sin(y xcos(y+x, f 3 (x,y sinβ y ( sin(u α du = a sinα sinβ(y x sinα x ( y x ( y + x 4a sinβ sin α Hence, for 0 P(L(ω = sin sin(α + β we get + sinβ sinα ((y x sin(y xcos(y+x α. f (0,α + f (α,π β + f 3 (π β,π πa sinα sinβ + a(sinα + sinβ + sin(α + β. ( b sin(α + β. We have P(L(ω = f (0,α + f (α,α + ϕ + f (π β φ,π β + f 3 (π β,π πa, sinα sinβ + a(sinα + sinβ + sin(α + β (3 where ϕ = arcsin sin(α + β α, φ = arcsin sin(α + β β. { } c min sin(α + β,asinβ, for which we have P(L(ω = πa sinα sinβ + a(sinα + sinβ + sin(α + β ( f (0,α + f (α,α + ϕ + (4 + f (π β φ,π β + f 3 (π β,π β + ϕ + f 3 (π φ,π, where ϕ = α + β π + arcsin, φ = arcsin α : c if sinβ sinα,we consider asinβ sin(α + β sin(α + β, so

6 6 Proc. of the Yerevan State Univ., Phys. and Math. Sci., 07, 5(3, p. 6. P(L(ω = πa sinα sinβ + a(sinα + sinβ + sin(α + β ( f (0,ϕ 3 + f (α φ 3,α + f (α,α + ϕ + f (π β φ,π β+ (5 + f 3 (π β,π β + ϕ + f 3 (π φ,π, where ϕ 3 = arcsin asinβ β, φ 3 = α + β π + arcsin asinβ ; sinα c if sinβ, we consider asinβ, then sin(α + β sin(α + β f (0,α + f (α,α + ϕ + f 3 (π φ,π P(L(ω = πa sinα sinβ + a(sinα + sinβ + sin(α + β. (6 { } d max sin(α + β,asinβ asinβ sin(α + β, P(L(ω = f (0,ϕ 3 + f (α φ 3,α + f (α,α + ϕ + f 3 (π φ,π πa. sinα sinβ + a(sinα + sinβ + sin(α + β (7 asinβ e a we have sin(α + β f (0,ϕ 3 + f 3 (π φ,π P(L(ω = πa sinα sinβ + a(sinα + sinβ + sin(α + β. (8 Obviousy, if > a, the probabiity P(L(ω is zero. Particuary, for reguar triange with a side a and α = β = 60, among a 5 subcases a e there are ony two cases, namey 0 sinα and sinα a and resut of Theorem coincides with the resut of [7] (Eqs. (4.3, (4.4 for a reguar triange. R E F E R E N C E S Received Santao L.A. Integra Geometry and Geometric Probabiity. Addision-Wesey, Schneider R., Wei W. Stochastic and Integra Geometry. Springer, Gasparyan A.G., Ohanyan V.K. Recognition of Trianges by Covariogram. // J. of Contemporary Mathematica Anaysis (Armenian Academy of Sciences, 03, v. 48, 3, p Harutyunyan H.S., Ohanyan V.K. Orientation-Dependent Section Distributions for Convex Bodies. // J. of Contemporary Mathematica Anaysis (Armenian Academy of Sciences, 04, v. 49, 3, p Gasparyan A.G., Ohanyan V.K. Orientation-Dependent Distribution of the Length of a Random Segment and Covariogram. // Journa of Contemporary Mathematica Anaysis (Armenian Academy of Sciences, 05, v. 50,, p Aharonyan N.G., Ohanyan V.K. Kinematic Measure of Intervas Lying in Domains. // J. of Contemporary Mathematica Anaysis (Armenian Academy of Sciences, 0, v. 46, 5, p Aharonyan N.G., Ohanyan V.K. Cacuation of Geometric Probabiities Using Covariogram of Convex Bodies. // Journa of Contemporary Mathematica Anaysis (Armenian Academy of Sciences, 08, v. 53, 3.