chord length distribution function for convex · pdf filesutra: international journal of...

Sutra: International Journal of Mathematical Science Education © Technomathematics Research Foundation

Vol. 4 No. 2, pp. 1 - 15, 2011

CHORD LENGTH DISTRIBUTION FUNCTION FOR CONVEX POLYGONS

H. S. HARUTYUNYAN

Department of Mathematics and Mechanics, Yerevan State University, Alex Manoogian, 1

0025 Yerevan, Armenia. [email protected].

V. K. OHANYAN∗

Department of Mathematics and Mechanics, Yerevan State University, Alex Manoogian, 1

0025 Yerevan, Armenia. [email protected].

Using the inclusion-exclusion principle and the Pleijel identity, an algorithm for calculation chord length distribution function for a bounded convex polygon is obtained. In the particular case an expression for the chord length distribution function for a rhombus is obtained.

Keywords : chord length distribution function; Pleijel identity; rhombus.

1. INTRODUCTION

LetG be thespace of linesg in the EuclideanplaneR2 , (p, φ)= the polar co‐ordinates ofthefootoftheperpendicular tog from theoriginO, be standardcoordinatesfora lineg ∈ G.

Letµ(·)standfor locallyfinitemeasureonG invariantwithrespect tothegroupof all Euclidean motions (translations and rotations). Itiswellknownthatthe elementof themeasureup to a constant factor has the followingform(see[1],[2]):

µ(dg)=dg =dp dφ,

wheredp isonedimensionalLebesguemeasure,whiledφ istheuniformmeasureontheunit circle.For eachboundedconvexdomainD thesetoflinesthat intersectD wedenoteby

[D]={g ∈ G : g ∩ D =∅}

∗ This research was partially supported by the State Committee of Science of the Republic of Armenia, Grant 11-1A-125.

1

2 Harutyunyan, Ohanyan

and we have (see [1], [2]):

µ([D]) = |∂D|,

where ∂D is the boundary of D and |∂D| stands for the length of ∂D.Let Ay

D be the set of lines that intersect D producing a chord χ(g) = g∩D of length

less than or equal to y:

AyD = {g ∈ [D] : |χ(g)| ≤ y}, y ∈ R.

Distribution function of the length of a random chord χ of D is defined as

F (y) =1

|∂D|µ(Ay

D) =1

|∂D|

∫∫Ay

D

dφ dp. (1.1)

Therefore, to obtain chord length distribution function for a bounded convex domain

D we have to calculate the integral in the right-hand side of (1.1). Explicit formulae

for the chord length distribution functions are known only for the cases of a disc, a

rectangle [3], and a regular polygon [5].

The main result of the paper is an algorithm for calculation the chord length dis-

tribution function for a bounded convex polygon. In particular, an expression for

the chord length distribution function for a rhombus is obtained.

The determination of the chord length distribution function has a long tradition of

application to collections of bounded convex bodies forming structures in metal and

ceramics. The series of formulae for chord length distribution functions may be of

use in finding suitable models when empirical distribution functions are given (see

[5], [6], [9] and [12]).

In [11] Sulanke proved that χ(g) = g ∩D is a measurable function on [D]. He also

proved that the function F (y) defined by formula (1.1) is a continuous function

with respect to y. In [8] Gates obtained some properties and inequalities for the

chord length distribution function for any bounded convex domain on the plane.

In particular, he proved that in the case of a bounded convex polygon the chord

length distribution function is expressed in terms of elementary functions.

2. THE CASE OF A CONVEX POLYGON

Let D be a convex bounded polygon in the plane and a1, a2,..., an be sides of D.Then

[D] =∪i<j

([ai] ∩ [aj ])

where [ai] ∩ [aj ] is the set of lines hitting both sides ai and aj of D.We write

F (y) =1

|∂D|∑i<j

∫∫{g∈[ai]∩[aj ]:|χ(g)|≤y}

dφdp =

Chord length distribution function... 3

=1

|∂D|

∑i<j

I∫∫

{g∈[ai]∩[aj ]:|χ(g)|≤y}dφdp+

∑i<j

II∫∫

{g∈[ai]∩[aj ]:|χ(g)|≤y}dφdp

,

where∑I

is over all pairs of nonparallel segments ai, aj ⊂ ∂D and∑II

is over all

pairs of parallel segments ai and aj ⊂ ∂D.In each of the integrals in the sum

∑i<j

Ione integration can be performed by passing

to the (|χ|, φ) coordinates. We have (see [2], page 157):

dg =sinα1 sinα2

| sin(α1 + α2)|d|χ| dφ, (2.1)

where α1 is the angle between ai and χ(g) = g ∩ D, while α2 is the angle between

aj and χ(g) (g ∈ [ai] ∩ [aj ]), α1 and α2 lie in one half-plane with respect to inside

of D.

3. PLEIJEL IDENTITY

Let D be a convex bounded polygon in the plane and a1,..., an be sides of D. Theso–called Pleijel identity for D is as follows (see [2], page 156):∫

[D]f(|χ(g)|) dg =

∫Gf ′(|χ|) · |χ| cotα1 cotα2 dg +

n∑i=1

∫ |ai|

0

f(u) du, (3.1)

where f(x) is a function with continuous first derivative f ′(x), α1 and α2 are the

angles between ∂D and g at the endpoints of χ(g) = g∩D which lie in one half-plane

with respect to the inside of D, |ai| is the length of ai, i = 1, ..., n.

It was shown by R. V. Ambartzumian in [2] (page 156) that the identity (3.1)

is useful to calculate chord length distribution function for the case of a bounded

convex polygon. If in (3.1) we formally put

fy(u) =

{0 if u ≤ y

1 if u > y

then the left-hand integral in (3.1) will equal

µ{g ∈ [D] : |χ(g)| > y},

i.e. the invariant measure of the set of chords of D whose length exceeds y. The

derivative of fy(u) should be replaced by Dirac’s δ–function concentrated at y (see

[2], page 156). Therefore, we obtain

[1− F (y)] |∂D| =∑i<j

I∫∫

[ai]∩[aj ]

δ(|χ| − y) · |χ| cotα1 cotα2 dg+

+∑i<j

II∫∫

[ai]∩[aj ]

δ(|χ| − y) · |χ| cotα1 cotα2 dg +n∑

i=1

(|ai| − y)+, (3.2)

where x+ = x if x > 0, and 0 otherwise.


For any continuous function f we have (see [10]):∫Rn

δ(x− y) f(x) dx = f(y). (3.3)

For each pair {ai, aj} ∈∑i<j

Iwe make the change of variables (p, φ) → (|χ|, φ) and

using (2.1) and (3.3) we obtain∫∫[ai]∩[aj ]

δ(|χ| − y) · |χ| cotα1 cotα2 dg =y

sin γij

∫Φij(y)

sinφ sin(γij − φ) dφ,

where γij is the angle between nonparallel sides ai and aj (or their continuations),

φ is the angle between direction φ and direction of segment ai i < j, and

Φij(y) = {φ : a chord joining ai and aj exists with direction φ and length y}.

Note, that Φij(y) is a subset in the space of directions in the plane, while in the

integral in the right hand side the corresponding Φij(y) is the set of angles, where

reference direction coincides with direction of segment ai, i < j and reference origin

coincides with the intersection of the lines containing ai and aj .

Moreover, for parallel sides ai and aj (i.e. ai, aj ∈∑i<j

II) we have

∑i<j

II∫∫

[ai]∩[aj ]

δ(|χ| − y) · |χ| cotα1 cotα2 dg =

= −∑i<j

IIIij(y)

∫δ(|χ| − y) · |χ|h(φχ) tan

2 φχdφχ

d|χ|d|χ| =

= −∑i<j

IIIij(y)h(φy) tan

2 φybij√

y2 − b2ij

,

where φy takes on two values arccosbijy or 2π−arccos

bijy , bij is the distance between

the parallel segments ai and aj (i.e. the distance between the lines containing ai and

aj), and I(A) is the indicator function of the event A, i.e. I(A) = 1 if A has occurred

and 0 otherwise, indicator Iij(y) = I(length of shortest chord hitting ai and aj ≤y ≤ length of longest chord hitting ai and aj). Further, h(φ) = bij is the height of

the maximal parallelogram with two sides equal to χ(φ) = g(φ)∩D, g(φ) ∈ [ai]∩[aj ](g(φ) is a line with φ-direction), and the other two sides lie on the parallel sides aiand aj ,

h(φχ) = h

(arccos

bij|χ(φ)|

)+ h

(2π − arccos

bij|χ(φ)|

).

Hence, h(·) = 0 if the parallelogram is empty.

For the value of φ such that |χ(φ)| = y we have h(φy) = h(φχ).


Therefore we obtain

F (y) = 1− 1n∑

i=1

|ai|

∑i<j

I y

sin γij

∫Φij(y)

sinφ sin(γij − φ) dφ−

−∑i<j

IIIij(y)h(φy)

√y2 − b2ij

bij+

n∑i=1

(|ai| − y)+

. (3.4)

In the case where ∂D contains no pairs of parallel sides, formula (3.4) coincides with

the expression given by R. V. Ambartzumian in [2], page 158.

It follows from (3.4) that to find distribution function F (y) we have to calculate

integrals of the form

1

sin γ

∫Φa,b(y)

sinφ sin(γ − φ) dφ

for any two nonparallel segments a and b (b ≤ a) with the angle γ between a and

b (or their continuations) and also calculate the second sum in (3.4) (for pairs of

parallel sides). Here and below Φa,b(y) is

Φa,b(y) = {φ : a chord joining a and b exists with direction φ and length y}.

4. THE DOMAIN Φa,b(y)

Consider segments a and b (b ≤ a), which have one common endpoint and make an

angle γ.

The line g with parameters (φ, p) intersect sides a and b, if

φ ∈(−π

2+ γ,

π

2

).

Without loss of generality we can assume, that the reference direction (X-axis)

coincides with the direction of the segment a and the origin O coincides with the

intersection of the lines containing a and b. Find the intersection point of the line

g = (φ, p) with the segment a. We have

x =p

cosφand y = 0.

Therefore,

g ∩ a =

x = p

cosφ

y = 0

0 ≤ p ≤ a cosφ.

Find the intersection point of the line g = (φ, p) with the segment b. We have{x cosφ+ y sinφ = p

y = tan γ xor

{x = p cos γk

cos(φ−γ)

y = p sin γcos(φ−γ)

.


Hence, we obtain

g ∩ b =

x = p cos γ

cos(φ−γ)

y = p sin γcos(φ−γ)

0 ≤ p ≤ b cos(φ− γ).

The line g which intersects sides a and b forms a chord χ of the length

|χ| = p sin γ

cosφ cos(φ− γ).

Therefore, we come to the following system of relations:0 ≤ p ≤ a cosφ

0 ≤ p ≤ b cos(φ− γ)

|χ| = p sin γcosφ cos(φ−γ) .

(4.1)

If a cosφ ≤ b cos(φ − γ), then tanφ ≥ a−b cos γb sin γ . Hence, φ ≥ arctan a−b cos γ

b sin γ .

It is not difficult to verify, that arctan a−b cos γb sin γ ∈

(−π

2 + γ, π2

). Since φ ∈(

−π2 + γ, arctan a−b cos γ

b sin γ

), then (4.1) has the following form:{

0 ≤ p ≤ b cos(φ− γ)


Moreover, we get

χmin(φ) = 0, χmax(φ) =b sin γ

cosφ.

The function χmax(φ) in the domain(−π

2 + γ, 0)decreasing, while in the domain(

0, arctan a−b cos γb sin γ

)increasing. Hence, we have

φmin = 0, χmax

(−π

2+ γ

)= b, χmax(φmin) = b sin γ and

χmax

(arctan

a− b cos γ

b sin γ

)=

√a2 + b2 − 2ab cos γ = c,

where c is the third side of the triangle made by a and b.

Now let φ ∈ (arctan a−b cos γb sin γ , π

2 ). In this case (4.1) can be rewritten in the form:{0 ≤ p ≤ a cos(φ))


Moreover, we get

χmin(φ) = 0, χmax(φ) =a sin γ

cos(φ− γ).


If arctan a−b cos γb sin γ > γ

(i.e. cos γ > b

a or γ < arccos ba

), than the function χmax(φ)

in the domain(arctan a−b cos γ

b sin γ , π2

)increasing,

χmax

(arctan

a− b cos γ

b sin γ− γ

)= c, χmax

(π2

)= a.

If arctan a−b cos γb sin γ ≤ γ

(i.e. cos γ ≤ b

a or γ ≥ arccos ba

), than the function χmax(φ)

in the domain(arctan a−b cos γ

b sin γ , γ)

decreasing, while in the domain(γ, π

2

)increas-

ing,

φmin = γ, χmax(φmin) = a sin γ, χmax

(arctan

a− b cos γ

b sin γ− γ

)= c, and

χmax

(π2

)= a.

Let γ < arccos ba . It is not difficult to verify that c < b, if cos γ > a

2b , or

γ < arccos a2b . From the other hand, arccos a

2b < arccos ba , if a > b

√2.

So, if a > b√2 and γ ≤ arccos a

2b , than

Φa,b(y) =

(−π

2 + γ, π2

), if y ∈ [0, b sin γ)(

−π2 + γ,− arccos b sin γ

y

)∪

∪(arccos b sin γ

y , π2

), if y ∈ [b sin γ, c)(


y

)∪

∪(γ + arccos a sin γ

y , π2

), if y ∈ [c, b)(

γ + arccos a sin γy , π

2

), if y ∈ [b, a]

∅, if y > a or y < 0

(4.2)

If a > b√2 and arccos a

2b < γ ≤ arccos ba , we have

Φa,b(y) =

(−π

2 + γ, π2

), if y ∈ [0, b sin γ)(


y

)∪

∪(arccos b sin γ

y , π2

), if y ∈ [b sin γ, b)(

arccos b sin γy , π

2

), if y ∈ [b, c)(


2

), if y ∈ [c, a]

∅, if y > a or y < 0

(4.3)

When a ≤ b√2, for every γ < arccos b

a , c < b, and Φa,b(y) is defined by (4.2).

Now let arccos ba ≤ γ < π

2 . It follows from the Sine theorem for the triangles, that

c ≥ a sin γ. Moreover, if a > b√2, arcsin b

a < arccos ba , so for every γ ≥ arccos b

a ,

a sin γ ≥ b.


Therefore, in the case a > b√2 and arccos b

a ≤ γ < arccos b2a we have

Φa,b(y) =

(−π

2 + γ, π2

), if y ∈ [0, b sin γ)(


y

)∪

∪(arccos b sin γ

y , π2

), if y ∈ [b sin γ, b)(


2

), if y ∈ [b, a sin γ)(

arccos b sin γy , γ − arccos a sin γ

y

)∪


y , π2

), if y ∈ [a sin γ, c)(


2

), if y ∈ [c, a)

∅, if y > a or y < 0

(4.4)

When a > b√2 and arccos b

2a ≤ γ < π2 we have

Φa,b(y) =

(−π

2 + γ, π2

), if y ∈ [0, b sin γ)(


y

)∪

∪(arccos b sin γ

y , π2

), if y ∈ [b sin γ, b)(


2

), if y ∈ [b, a sin γ)(


y

)∪


y , π2

), if y ∈ [a sin γ, a)(


y

), if y ∈ [a, c)

∅, if y > c or y < 0

(4.5)

Let a ≤ b√2 and arccos b

a ≤ γ < π2 . If in addition γ ≤ arccos a

2b , than we have

c < b. So, in this case we have

Φa,b(y) =

(−π

2 + γ, π2

), if y ∈ [0, b sin γ)(


y

)∪

∪(arccos b sin γ

y , π2

), if y ∈ [b sin γ, a sin γ)(


y

)∪

∪(arccos b sin γ

y , γ − arccos a sin γy

)∪


y , π2

)if y ∈ [a sin γ, c)(


y

)∪


y , π2

)if y ∈ [c, b)(


2

), if y ∈ [b, a)

∅, if y > a or y < 0

(4.6)


If a ≤ b√2 and arccos a

2b ≤ γ ≤ arcsin ba , than a sin γ ≤ b and c ≥ b, so we have

two cases: c ≤ a and c > b. By that mean we split the condition a ≤ b√2 into

two conditions: a ≤ b√5

2 and b√5

2 ≤ a ≤ b√2. Now let a ≤ b

√5

2 , arccos a2b ≤ γ <

arccos b2a . In this case we have b sin γ ≤ a sin γ ≤ b ≤ c ≤ a, so

Φa,b(y) =

(−π

2 + γ, π2

), if y ∈ [0, b sin γ)(


y

)∪

∪(arccos b sin γ

y , π2



y

)∪

∪(arccos b sin γ


)∪


y , π2

)if y ∈ [a sin γ, b)(


y

)∪


y , π2

)if y ∈ [b, c)(


2

), if y ∈ [c, a)

∅, if y > a or y < 0

(4.7)

When a ≤ b√5

2 and arccos b2a ≤ γ < arcsin b

a , we have b sin γ ≤ a sin γ ≤ b ≤ a ≤c, so

Φa,b(y) =

(−π

2 + γ, π2

), if y ∈ [0, b sin γ)(


y

)∪

∪(arccos b sin γ

y , π2



y

)∪

∪(arccos b sin γ


)∪


y , π2

)if y ∈ [a sin γ, b)(


y

)∪


y , π2

)if y ∈ [b, a)(


y

), if y ∈ [a, c)

∅, if y > c or y < 0

(4.8)

In case of a ≤ b√5

2 and arcsin ba < γ < π

2 , Φa,b(y) is defined by (4.5). Let b√5

2 ≤a ≤ b

√2. It is not difficult to verify that in this case when arccos a

2b ≤ γ < arcsin ba ,

Φa,b(y) is defined by the formula (4.7), and when arcsin ba ≤ γ < arccos b

2a , Φa,b(y)

is defined by (4.4), while in the case of arccos b2a ≤ γ < π

2 it’s defined by (4.5).

Consider γ ≥ π2 (c > b, c > a). In this case the function b sin γ

cosφ in the

domain(−π

2 , arctana−b cos γb sin γ

)increasing, while function a sin γ

cos(φ−γ) in the domain


(arctan a−b cos γ

b sin γ , π2

)decreasing, so

Φa,b(y) =

(−π

2 + γ, π2

), if y ∈ [0, b)(


2

), if y ∈ [b, a)(


y

), if y ∈ [a, c)

∅, if y > c or y < 0

(4.9)

Finally, we obtain the following table

Cases Φa,b(y)

a > b√2 and γ ≤ arccos a

2b , a ≤ b√2 and γ < arccos b

a (4.2)

a > b√2 and arccos a

2b < γ ≤ arccos ba (4.3)

a > b√2 and arccos b

a ≤ γ < arccos b2a , (4.4)

b√5

2 ≤ a ≤ b√2 and arcsin b

a ≤ γ < arccos b2a

a > b√5

2 and arccos b2a ≤ γ < π

2 ,(4.5)

a ≤ b√5

2 and arcsin ba < γ < π

2

a ≤ b√2 and γ ≤ arccos a

2b (4.6)

a ≤ b√5

2 and arccos a2b ≤ γ < arccos b

2a ,(4.4)

b√5

2 ≤ a ≤ b√2 and arccos a

2b ≤ γ < arcsin ba

a ≤ b√5

2 and arccos b2a ≤ γ < arcsin b

a (4.8)

π2 ≤ γ < π (4.9)

Table 1

When segments a and b have no common endpoint, we put them so, that the

intersection point of lines containing a and b coincide with the origin, and segment

a lie on X-axis. Denote by a′ and b′ the segments containing a and b, where one of

the endpoints of a′ and b′ coincide with the origin, and another endpoint coincides

with the endpoint of a and b correspondingly. Denote a′′ = a′ a, b′′ = b′ b. It is not

difficult to verify that

I[a]∩[b](g) = I[a′]

∩[b′](g)− I[a′]

∩[b′′](g)− I[a′′]

∩[b](g) + I[a′′]

∩[b′′](g) ,

where I is the characteristic function. Here segments a′ and b′, a′ and b′′, a′′ and

b′, a′′ and b′′ have common endpoint, so we can use the table above.

In fact we obtain that for every convex polygon D we can find the domains

Φij(y) and calculate the integrals in (3.4).


5. CHORD LENGTH DISTRIBUTION FUNCTION FOR

RHOMBUS

Consider the special case of (3.4) when D is a rhombus with side a and angle γ(γ ≤ π

2

):

F (y) = 1− y2a sin γ

[∫Φ12(y)

sinφ sin(γ − φ) dφ+∫Φ14(y)

sinφ sin(π − γ − φ) dφ]

+

√y2−a2 sin2 γ

2ay

[I(a sin γ ≤ y ≤ 2a sin γ

2 )(a(1− cos γ)−

√y2 − a2 sin2 γ

)+I(a sin γ ≤ y ≤ a)

(a(1− cos γ) +


)+I(a ≤ y ≤ 2a cos

γ

2)

(a(1 + cos γ)−


)]− (a− y)+

a. (5.1)

From Table 1 for Φ12(y) we have 2 cases: γ ≤ π3 , and π

3 < γ ≤ π2 , which are

defined by (4.6), (4.8) correspondingly. Let γ ≤ π3 . We have

Φ12(y) =

(−π

2 + γ, π2

), if y ∈ [0, a sin γ)(

−π2 + γ,− arccos a sin γ

y

)∪

∪(arccos a sin γ


)∪


y , π2

)if y ∈ [a sin γ, 2a sin γ

2 )(−π

2 + γ,− arccos a sin γy

)∪


y , π2

)if y ∈ [2a sin γ

2 , a)

∅, if y > a or y < 0

Φ14(y) is defined by (4.9):

Φ14(y) =

(π2 − γ, π

2

), if y ∈ [0, a)(

arccos a sin γy , π − γ − arccos a sin γ

y

), if y ∈ [a, 2a cos γ

2 )

∅, if y > 2a cos γ2 or y < 0

Calculating integrals in (5.1) we obtain

F (y) =

0, if y < 0y2a

[1 +

(π2 − γ

)cot γ

], if y ∈ [0, a sin γ)

y2a

[1−

(π2 + γ − 2 arcsin a sin γ

y

)cot γ

]+

+

√y2−a2 sin2 γ

y , if y ∈ [a sin γ, 2a sin γ2 )

y4a

[3 +

(2 arcsin a sin γ

y − 3γ)cot γ

]+

+

√y2−a2 sin2 γ

2y , if y ∈ [2a sin γ2 , a)

1 + y4a

[−1 +

(γ − 2 arcsin a sin γ

y

)cot γ

]+

+

√y2−a2 sin2 γ

2y if y ∈ [a, 2a cos γ2 )

1, if y ≥ 2a cos γ2

(5.2)


It is not difficult to verify that function defined by (5.2) is a continuous function.

Now let π3 < γ ≤ π

2 . From (4.8) we have

Φ12(y) =

(−π

2 + γ, π2

), if y ∈ [0, a sin γ)(

−π2 + γ,− arccos a sin γ

y

)∪

∪(arccos a sin γ


)∪


y , π2

)if y ∈ [a sin γ, a)(

arccos a sin γy , γ − arccos a sin γ

y

)if y ∈ [a, 2a sin γ

2 )

∅, if y > a or y < 0

As in previous case Φ14(y) is defined by (4.9). Finally we have

F (y) =

0, if y < 0y2a

[1 +

(π2 − γ

)cot γ

], if y ∈ [0, a sin γ)

y2a

[1−


y

)cot γ

]+

+

√y2−a2 sin2 γ

y , if y ∈ [a sin γ, a)

1− y2a

[1 +

(π2 − γ

)cot γ

]+

√y2−a2 sin2 γ

y , if y ∈ [a, 2a sin γ2 )

1 + y4a

[−1 +


y

)cot γ

]+

+

√y2−a2 sin2 γ

2y if y ∈ [2a sin γ2 , 2a cos

γ2 )

1, if y ≥ 2a cos γ2

(5.3)

In (5.3) function F (y) is continuous.

Thus, in terms of elementary functions we find the elementary expression of

chord length distribution for rhomb. It is defined by formulae (5.2) and (5.3). If we

denote by f(y) = F ′(y) the chord length density function, then for γ ≤ π3 and for

π3 < γ ≤ π

2 we have correspondingly

f(y) =

0, if y < 0 or y ≥ 2a cos γ2

12a

[1 +

(π2 − γ

)cot γ

], if y ∈ [0, a sin γ)

12a

[1−


y

)cot γ

]+

+a2 sin2 γ+y2 cos γ

y2√

y2−a2 sin2 γ, if y ∈ [a sin γ, 2a sin γ

2 )

14a

[3 +

(2 arcsin a sin γ

y − 3γ)cot γ

]+

+ a2 sin2 γ−y2 cos γ

2y2√

y2−a2 sin2 γ, if y ∈ [2a sin γ

2 , a)

14a

[−1 +


y

)cot γ

]+

+ a2 sin2 γ+y2 cos γ

2y2√

y2−a2 sin2 γ, if y ∈ [a, 2a cos γ

2 )

(5.4)


f(y) =

0, if y < 0 or y ≥ 2a cos γ2

12a

[1 +

(π2 − γ

)cot γ

], if y ∈ [0, a sin γ)

12a

[1−


y

)cot γ

]+

+a2 sin2 γ+y2 cos γ

y2√

y2−a2 sin2 γ, if y ∈ [a sin γ, a)

− 12a

[1 +

(π2 − γ

)cot γ

]+

+ a sin2 γ

y2√

y2−a2 sin2 γ, if y ∈ [a, 2a sin γ

2 )

14a

[−1 +


y

)cot γ

]+

+ a2 sin2 γ+y2 cos γ

2y2√

y2−a2 sin2 γ, if y ∈ [2a sin γ

2 , 2a cosγ2 ),

(5.5)

The graphs of chord length distribution and density function for rhombus with

side a = 1 and angles γ = π6 and γ = 5π

12 see in Figures 1 and 2.

References

[1] R. Schneider and W. Weil(1990). Stochastic and Integral Geometry, Springer Verlag,Berlin.

[2] Ambartzumian, R. (1990) Factorization Calculus and Geometric Probability, Cam-bridge University Press, Cambridge.

[3] Gille, W. (1988). The chord length distribution of parallelepipeds with their limitingcases, Exp. Techn. Phys. 36, 197–208.

[4] Aharonyan, N. and Ohanyan, V. (2005). Chord length distribution functions for poly-gons, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sci-ences). 40, no. 4, 43–56.

[5] Harutyunyan, N. and Ohanyan, V. (2009). Chord length distribution function for reg-ular polygon, J. Appl. Prob. 42, no. 2, 358–366.

[6] Stoyan D. and Stoyan,H. (1994). Fractals, Random Shapes and Point Fields, JohnWiley& Sons, Chichester, New York, Brisbane, Toronto, Singapore.

[7] Gates J. (1982). Recognition of triangles and quadrilaterals by chord length distribution,J. Appl. Prob. 19, 873–879.

[8] Gates J.(1987). Some properties of chord length distributions, J. Appl. Prob. 24 863–874.

[9] Gille W., Aharonyan N. G. and Haruttyunyan H. S.(2009). Chord length distributionof pentagonal and hexagonal rods: relation to small-angle scattering. In Journal ofApplied Cristallography 326–328.

[10] Shilov, G. (1984). Mathematical Analysis, (second special course), 2nd edn, Moscow,Moscow state University.

[11] R. Sulanke. (1961). Die Verteilung der Sehnenlangen an Ebenen und raumlichen Fig-uren, Math. Nachr.,, 23, 51 – 74.

[12] V. K. Ohanyan and N. G. Aharonyan (2009). Tomography of bounded convex domains,Sutra: International Journal of Mathematical Science Education, 2, no. 1, 1 - 12,


Fig. 1. Chord length distribution and density functions for rhombus with a = 1 and γ = π6 .


Fig. 2. Chord length distribution and density functions for rhombus with a = 1 and γ = 5π12 .

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