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 lines g in the Euclideanplane R2 , (p, φ) = the polar co‐
ordinates ofthefootoftheperpendicular tog from theorigin O, be standard
coordinatesfora lineg ∈ G.
Let µ(·)standfor locallyfinitemeasure on G invariantwithrespect tothe
group of all Euclidean motions (translations and rotations). Itis wellknownthat
the elementof the measureup to a constant factor has the followingform(see[1],
[2]):
µ(dg)=dg =dp dφ,
wheredp isonedimensionalLebesguemeasure,whiledφ istheuniformmeasureon
the unit circle.
For eachboundedconvexdomainD the setoflinesthat 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 AyD 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
∫∫
1
1
y
µ(AD ) =
dφ dp.
F (y) =
|∂D|
|∂D|
Ay
D
(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 distribution 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] =
([ai ] ∩ [aj ])
i<j
where [ai ] ∩ [aj ] is the set of lines hitting both sides ai and aj of D.
We write
∫∫
1 ∑
F (y) =
dφ dp =
|∂D| i<j
{g∈[ai ]∩[aj ]:|χ(g)|≤y}
Chord length distribution function...
3


∫∫
∑II ∫ ∫
1 ∑I
=
dφ dp +
dφ dp ,
|∂D| i<j
{g∈[ai ]∩[aj ]:|χ(g)|≤y}
{g∈[ai ]∩[aj ]:|χ(g)|≤y}
i<j
∑I
∑II
where
is over all pairs of nonparallel segments ai , aj ⊂ ∂D and
is over all
pairs of parallel segments ai and aj ⊂ ∂D.
∑I
In each of the integrals in the sum
one integration can be performed by passing
i<j
to the (|χ|, φ) coordinates. We have (see [2], page 157):
dg =
sin α1 sin α2
d|χ| dφ,
| sin(α1 + α2 )|
(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. The
so–called Pleijel identity for D is as follows (see [2], page 156):
∫
∫
n ∫ |ai |
∑
f (|χ(g)|) dg =
f ′ (|χ|) · |χ| cot α1 cot α2 dg +
f (u) du,
(3.1)
G
[D]
i=1
0
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
{
0
if
u≤y
fy (u) =
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
∑I ∫ ∫
δ(|χ| − y) · |χ| cot α1 cot α2 dg+
[1 − F (y)] |∂D| =
∑II
+
i<j
∫∫
i<j
[ai ]∩[aj ]
δ(|χ| − y) · |χ| cot α1 cot α2 dg +
[ai ]∩[aj ]
where x+ = x if x > 0, and 0 otherwise.
n
∑
i=1
(|ai | − y)+ ,
(3.2)
4
Harutyunyan, Ohanyan
For any continuous function f we have (see [10]):
∫
δ(x − y) f (x) dx = f (y).
For each pair {ai , aj } ∈
∑I
(3.3)
Rn
we make the change of variables (p, φ) → (|χ|, φ) and
i<j
using (2.1) and (3.3) we obtain
∫∫
δ(|χ| − y) · |χ| cot α1 cot α2 dg =
[ai ]∩[aj ]
y
sin γij
∫
sin φ sin(γij − φ) dφ,
Φij (y)
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 .
∑ II
Moreover, for parallel sides ai and aj (i.e. ai , aj ∈
) we have
i<j
∑II ∫ ∫
i<j
δ(|χ| − y) · |χ| cot α1 cot α2 dg =
[ai ]∩[aj ]
∫
∑II
dφχ
=−
Iij (y) δ(|χ| − y) · |χ| h(φχ ) tan2 φχ
d|χ| =
d|χ|
i<j
∑II
bij
=−
Iij (y) h(φy ) tan2 φy √
,
y 2 − b2ij
i<j
b
b
where φy takes on two values arccos yij or 2 π−arccos yij , 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 ai
and aj ,
)
(
)
(
bij
bij
+ h 2 π − arccos
.
h(φχ ) = h arccos
|χ(φ)|
|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 ).
Chord length distribution function...
Therefore we obtain
F (y) = 1 − ∑
n

∑I

1
|ai |
i<j
y
sin γij
i=1
−
∑II
∫
sin φ sin(γij − φ) dφ−
Φij (y)
√
Iij (y) h(φy )
5
y 2 − b2ij
i<j
bij
+
n
∑

(|ai | − y)+  .
(3.4)
i=1
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 φ sin(γ − φ) dφ
sin γ Φa,b (y)
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
( π
π)
φ ∈ − + γ,
.
2
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
p
x=
and y = 0.
cos φ
Therefore,
g∩a=


x =

p
cos φ
y=0


0 ≤ p ≤ a cos φ.
Find the intersection point of the line g = (φ, p) with the segment b. We have
{
{
p cos γk
x = cos(φ−γ)
x cos φ + y sin φ = p
or
.
p sin γ
y = tan γ x
y = cos(φ−γ)
6
Harutyunyan, Ohanyan
Hence, we obtain

p cos γ


x = cos(φ−γ)
p sin γ
g ∩ b = y = cos(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
.
cos φ cos(φ − γ)
Therefore, we come to the following system of relations:



0 ≤ p ≤ a cos φ
0 ≤ p ≤ b cos(φ − γ)


p sin γ
|χ| =
cos φ cos(φ−γ) .
If a cos φ ≤ b cos(φ − γ), then tan φ ≥
a−b cos γ
b sin γ .
a−b cos γ
b sin γ
(4.1)
cos γ
Hence, φ ≥ arctan a−b
b sin γ .
( π
)
∈ − 2 + γ, π2 . Since φ ∈
It is not difficult to verify, that arctan
(
)
cos γ
− π2 + γ, arctan a−b
, then (4.1) has the following form:
b sin γ
{
0 ≤ p ≤ b cos(φ − γ)
|χ| =
p sin γ
cos φ cos(φ−γ) .
Moreover, we get
χmin (φ) = 0,
χmax (φ) =
b sin γ
.
cos φ
( π
)
The
function
χ
(φ)
in
the
domain
−
+
γ,
0
decreasing, while in the domain
max
2
)
(
a−b cos γ
increasing. Hence, we have
0, arctan b sin γ
φmin = 0,
( π
)
χmax − + γ = b,
2
χmax (φmin ) = b sin γ
and
(
) √
a − b cos γ
χmax arctan
= a2 + b2 − 2ab cos γ = c,
b sin γ
where c is the third side of the triangle made by a and b.
cos γ π
Now let φ ∈ (arctan a−b
b sin γ , 2 ). In this case (4.1) can be rewritten in the form:
{
0 ≤ p ≤ a cos(φ))
|χ| =
p sin γ
cos φ cos(φ−γ) .
Moreover, we get
χmin (φ) = 0,
χmax (φ) =
a sin γ
.
cos(φ − γ)
Chord length distribution function...
7
)
(
cos γ
b
If arctan a−b
or γ < arccos ab , than the function χmax (φ)
b sin γ( > γ i.e. cos γ >
a
)
cos γ π
in the domain arctan a−b
increasing,
b sin γ , 2
)
(
(π)
a − b cos γ
χmax arctan
− γ = c,
χmax
= a.
b sin γ
2
(
)
cos γ
If arctan a−b
≤ γ i.e. cos γ ≤ ab or γ ≥ arccos ab , than the function χmax (φ)
b sin γ
(
)
(
)
cos γ
in the domain arctan a−b
decreasing, while in the domain γ, π2 increasb sin γ , γ
ing,
(
)
a − b cos γ
φmin = γ, χmax (φmin ) = a sin γ, χmax arctan
− γ = c, and
b sin γ
χmax
(π)
2
= a.
Let γ < arccos ab . It is not difficult to verify that c < b, if√ cos γ >
a
a
γ < arccos 2b
. From
the other hand, arccos 2b
< arccos ab , if a > b 2.
√
a
So, if a > b 2 and γ ≤ arccos 2b , than
(
)

if y ∈ [0, b sin γ)
− π2 + γ, π2 ,


(
)


b sin γ
π

− 2 + γ, − arccos y
∪



(
)


b
sin
γ
π

∪ arccos y , 2 ,
if y ∈ [b sin γ, c)


(
)
γ
Φa,b (y) =
− π2 + γ, − arccos b sin
y) ∪

(


γ π

∪ γ + arccos a sin
if y ∈ [c, b)

y , )2 ,


(


a sin γ π

if y ∈ [b, a]

 γ + arccos y , 2 ,


∅,
if y > a or y < 0
√
a
If a > b 2 and arccos 2b
< γ ≤ arccos ab , we have
(
)
π
π

+
γ,
,
if y ∈ [0, b sin γ)
−

2
2
(
)


b
sin
γ

∪
 − π2 + γ, − arccos y


(
)


∪ arccos b sin γ , π ,
if y ∈ [b sin γ, b)
y
)2
Φa,b (y) = (
b
sin
γ
π

arccos y , 2 ,
if y ∈ [b, c)



(
)


γ π

γ + arccos a sin
if y ∈ [c, a]

y , 2 ,



∅,
if y > a or y < 0
a
2b ,
or
(4.2)
(4.3)
√
When a ≤ b 2, for every γ < arccos ab , c < b, and Φa,b (y) is defined by (4.2).
Now let arccos ab ≤ γ < π2 . It
√follows from the Sine theorem for the triangles, that
c ≥ a sin γ. Moreover, if a > b 2, arcsin ab < arccos ab , so for every γ ≥ arccos ab ,
a sin γ ≥ b.
8
Harutyunyan, Ohanyan
√
b
Therefore, in the case a > b 2 and arccos ab ≤ γ < arccos 2a
we have
(
)
π
π

+
γ,
,
if y ∈ [0, b sin γ)
−

2
2
(
)


b sin γ
π

∪
 − 2 + γ, − arccos y


)
(


b sin γ π

if y ∈ [b sin γ, b)
∪ arccos y , 2 ,



(
)


b
sin
γ
π
 arccos
if y ∈ [b, a sin γ)
y , 2 ,
)
Φa,b (y) = (
b sin γ
a sin γ

arccos y , γ − arccos y
∪


 (
)


a
sin
γ
π

if y ∈ [a sin γ, c)
∪ γ + arccos y , 2 ,


(
)


γ π

if y ∈ [c, a)
γ + arccos a sin

y , 2 ,



∅,
if y > a or y < 0
√
b
When a > b 2 and arccos 2a
≤ γ < π2 we have
(
)

− π2 + γ, π2 ,


(
)


b sin γ
π

−
+
γ,
−
arccos
∪



(2
) y


b
sin
γ
π

∪ arccos y , 2 ,



)
(

 arccos b sin γ , π ,
y
2
)
Φa,b (y) = (
b sin γ
a sin γ

arccos
,
γ
−
arccos
∪

y
y


(
)


γ π

∪ γ + arccos a sin

y , 2 ,


(
)


b sin γ
a sin γ

arccos
,
γ
−
arccos
,

y
y



∅,
(4.4)
if y ∈ [0, b sin γ)
if y ∈ [b sin γ, b)
if y ∈ [b, a sin γ)
(4.5)
if y ∈ [a sin γ, a)
if y ∈ [a, c)
if y > c
√
Let a ≤ b 2 and arccos ab ≤ γ < π2 . If in addition
c < b. So, in this case we have
(
)

if
− π2 + γ, π2 ,


(
)


b sin γ
π

− 2 + γ, − arccos y
∪



(
)


b
sin
γ
π

∪ arccos y , 2 ,
if



(
)


γ
π

∪
+ γ, − arccos b sin

y
 −

(2
)


b
sin
γ
a sin γ
∪ arccos
,
γ
−
arccos
∪
y
y
(
)
Φa,b (y) =
γ π

∪ γ + arccos a sin
if

y , 2 )
(



b
sin
γ
π
 − + γ, − arccos

y) ∪


(2


γ π

∪ γ + arccos a sin
if

y , )2


(


a sin γ π

γ + arccos y , 2 ,
if




∅,
if
or
y<0
a
γ ≤ arccos 2b
, than we have
y ∈ [0, b sin γ)
y ∈ [b sin γ, a sin γ)
y ∈ [a sin γ, c)
y ∈ [c, b)
y ∈ [b, a)
y > a or
y<0
(4.6)
Chord length distribution function...
9
√
a
If a ≤ b 2 and arccos 2b
≤ γ ≤ arcsin ab , 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
√
b 5
b 5
b 5
a
two conditions: a ≤ 2 and 2 ≤ a ≤ b 2. Now let a ≤ 2 , arccos 2b ≤ γ <
b
arccos 2a
. In this case we have b sin γ ≤ a sin γ ≤ b ≤ c ≤ a, so
(
)

if y ∈ [0, b sin γ)
− π2 + γ, π2 ,


(
)


b
sin
γ
π

− 2 + γ, − arccos y
∪



(
)


γ π

arccos b sin
if y ∈ [b sin γ, a sin γ)

y , 2 ,
∪

(
)


b
sin
γ
π

∪
− 2 + γ, − arccos y



(
)


∪ arccos b sin γ , γ − arccos a sin γ ∪
y
y
(
)
(4.7)
Φa,b (y) =
a sin γ π

∪
γ
+
arccos
,
if y ∈ [a sin γ, b)

y
2


(
)


b sin γ
a sin γ

arccos
,
γ
−
arccos
∪

y
y
 (

)


a
sin
γ
π

∪ γ + arccos y , 2
if y ∈ [b, c)



)
(

γ π

if y ∈ [c, a)
γ + arccos a sin

y , 2 ,



∅,
if y > a or y < 0
When a ≤
c, so
√
b 5
2
b
and arccos 2a
≤ γ < arcsin ab , we have b sin γ ≤ a sin γ ≤ b ≤ a ≤
(
)
π
π

if

(− 2 + γ, 2 ,
)


b sin γ
π

∪
 − 2 + γ, − arccos y


)
(


b sin γ π

if
∪ arccos y , 2 ,



(
)


b
sin
γ

− π2 + γ, − arccos y
∪



(
)


∪ arccos b sin γ , γ − arccos a sin γ ∪
y
y
(
)
Φa,b (y) =
a sin γ π

∪
γ
+
arccos
,
if

y
2


(
)


γ
a sin γ

arccos b sin
∪

y , γ − arccos
y


(
)


a
sin
γ
π

∪ γ + arccos y , 2
if



(
)


γ
a sin γ

arccos b sin
, if

y , γ − arccos
y



∅,
if
√
y ∈ [0, b sin γ)
y ∈ [b sin γ, a sin γ)
y ∈ [a sin γ, b)
(4.8)
y ∈ [b, a)
y ∈ [a, c)
y>c
or
y<0
√
b 5
b 5
b
π
In case
√ of a ≤ 2 and arcsin a < γ < 2 , Φa,b (y) is defined bya(4.5). Let 2 b≤
a ≤ b 2. It is not difficult to verify that in this case when arccos 2b ≤ γ < arcsin a ,
b
, Φa,b (y)
Φa,b (y) is defined by the formula (4.7), and when arcsin ab ≤ γ < arccos 2a
b
π
is defined by (4.4), while in the case of arccos 2a ≤ γ < 2 it’s defined by (4.5).
sin γ
Consider γ ≥ π2 (c > b, c > a). In this case the function bcos
φ in the
(
)
a sin γ
a−b cos γ
π
domain − 2 , arctan b sin γ
increasing, while function cos(φ−γ) in the domain
10
(
Harutyunyan, Ohanyan
cos γ π
arctan a−b
b sin γ , 2
)
decreasing, so
(
)
π
π

+
γ,
,
if
−

2
2
(
)


b
sin
γ
π
 arccos
if
y , 2 ,
)
Φa,b (y) = (
b sin γ
a sin γ

arccos y , γ − arccos y
, if




∅,
if
y ∈ [0, b)
y ∈ [b, a)
(4.9)
y ∈ [a, c)
y>c
or
y<0
Finally, we obtain the following table
Cases
√
√
a ≤ b 2 and γ < arccos ab
a > b 2 and γ ≤ arccos
√
a
a > b 2 and arccos 2b
< γ ≤ arccos ab
√
b
,
a > b 2 and arccos ab ≤ γ < arccos 2a
√
√
b 5
b
b
2 ≤ a ≤ b 2 and arcsin a ≤ γ < arccos 2a
a
2b ,
√
b 5
2
√
b 5
≤ 2
a>
a
b
and arccos 2a
≤γ<
and arcsin ab < γ <
√
a
a ≤ b 2 and γ ≤ arccos 2b
a≤
√
b 5
2
a≤
√
b 5
2
√
b 5
2
π
2,
b
and arccos 2a
≤ γ < arcsin ab
≤γ<π
(4.2)
(4.3)
(4.4)
(4.5)
π
2
a
b
and arccos 2b
≤ γ < arccos 2a
,
√
a
≤ a ≤ b 2 and arccos 2b
≤ γ < arcsin ab
π
2
Φa,b (y)
(4.6)
(4.4)
(4.8)
(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).
Chord length distribution function...
11
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
π
γ≤ 2 :
[∫
]
∫
y
F (y) = 1 − 2a sin
sin
φ
sin(γ
−
φ)
dφ
+
sin
φ
sin(π
−
γ
−
φ)
dφ
γ
Φ12 (y)
Φ14 (y)
√
[
(
)
√
y 2 −a2 sin2 γ
γ
2 − a2 sin2 γ
+
I(a
sin
γ
≤
y
≤
2a
sin
)
a(1
−
cos
γ)
−
y
2ay
2
(
)
√
+I(a sin γ ≤ y ≤ a) a(1 − cos γ) + y 2 − a2 sin2 γ
(
)]
√
γ
(a − y)+
.
(5.1)
+I(a ≤ y ≤ 2a cos ) a(1 + cos γ) − y 2 − a2 sin2 γ
−
2
a
From Table 1 for Φ12 (y) we have 2 cases: γ ≤ π3 , and π3 < γ ≤ π2 , which are
defined by (4.6), (4.8) correspondingly. Let γ ≤ π3 . We have
(
)

if y ∈ [0, a sin γ)
− π2 + γ, π2 ,


(
)


a sin γ
π

+ γ, − arccos y
∪

 −

)
(2


γ
a
sin
γ
∪ arccos
, γ − arccos a sin
∪

y
y

 (
)
γ π
Φ12 (y) = ∪ γ + arccos a sin
if y ∈ [a sin γ, 2a sin γ2 )
y , 2

(
)


γ

− π2 + γ, − arccos a sin
∪

y)


(


a sin γ π

if y ∈ [2a sin γ2 , a)

∪ γ + arccos y , 2


∅,
if y > a or y < 0
Φ14 (y) is defined by (4.9):
(
)
π
π

if y ∈ [0, a)

(2 − γ, 2 ,
)
a sin γ
a sin γ
Φ14 (y) =
arccos y , π − γ − arccos y
, if y ∈ [a, 2a cos γ2 )


∅,
if y > 2a cos γ2 or y < 0
Calculating integrals in (5.1) we obtain


0,
if



[
(π
)
]

y

if


2a [1 + (2 − γ cot γ ,
]
)


y
a sin γ
π


cot γ +

2a 1 − 2 + γ − 2 arcsin
y

√


2
2
2

y −a sin γ

+
,
if


y(
)
]
 [
a sin γ
y
F (y) = 4a 3 + 2 arcsin y − 3γ cot γ +
√



y 2 −a2 sin2 γ


+
,
if

[2y
(
)
]


y
a
sin
γ


1 + 4a −1 + γ − 2 arcsin y
cot γ +


√


2
2
2

y −a sin γ


+
if

2y



1,
if
y<0
y ∈ [0, a sin γ)
y ∈ [a sin γ, 2a sin γ2 )
y ∈ [2a sin γ2 , a)
y ∈ [a, 2a cos γ2 )
y ≥ 2a cos γ2
(5.2)
12
Harutyunyan, Ohanyan
It is not difficult to verify that function defined by (5.2) is a continuous function.
Now let π3 < γ ≤ π2 . From (4.8) we have
(
)

if y ∈ [0, a sin γ)
− π2 + γ, π2 ,


(
)


a sin γ
π

+ γ, − arccos y
∪

 −

)
(2


a sin γ
a
sin
γ
∪ arccos
,
γ
−
arccos
∪
y
y
(
)
Φ12 (y) =
γ π

∪ γ + arccos a sin
if y ∈ [a sin γ, a)

y , 2
(

)


a
sin
γ
a
sin
γ

arccos y , γ − arccos y
if y ∈ [a, 2a sin γ2 )




∅,
if y > a or y < 0
As in previous case Φ14 (y) is defined by (4.9). Finally we have


0,
if y < 0


[
(π
)
]

y


if y ∈ [0, a sin γ)

2a [1 + (2 − γ cot γ ,
)
]



a sin γ
y
π

cot γ +

2a 1 − 2 + γ − 2 arcsin
y


√

2 −a2 sin2 γ

y


,
if y ∈ [a sin γ, a)

+
[y ( π
)
]
y
1 + 2 − γ cot γ
F (y) = 1 −
√ 2a


y 2 −a2 sin2 γ


,
if y ∈ [a, 2a sin γ2 )
+
y

[
(
)
]


y
γ


1 + 4a
−1 + γ − 2 arcsin a sin
cot γ +

y

√


2
2
2
y −a sin γ


+
if y ∈ [2a sin γ2 , 2a cos γ2 )

2y



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
π
π
3 < γ ≤ 2 we have correspondingly


if y < 0 or y ≥ 2a cos γ2
0,


[
(
)
]

1
π

if y ∈ [0, a sin γ)


2a [1 + (2 − γ cot γ ,
)
]


a
sin
γ

1
π

cot γ +

2a 1 − 2 + γ − 2 arcsin
y


2
2
2

a sin γ+y cos γ


if y ∈ [a sin γ, 2a sin γ2 )
+ y2 √y2 −a2 sin2 γ ,
[
(
)
]
f (y) =
a sin γ
1
− 3γ cot γ +

4a 3 + 2 arcsin
y



2
2
2

a
sin
γ−y
cos
γ

+ 2√ 2 2 2 ,
if y ∈ [2a sin γ2 , a)


2y
y −a
sin γ

[
(
)
]


a sin γ
1


cot γ +

4a −1 + γ − 2 arcsin
y


2
2
2

a
sin
γ+y
cos
γ

+ 2 √ 2 2 2 ,
if y ∈ [a, 2a cos γ2 )
2y
y −a sin γ
(5.4)
Chord length distribution function...


0,
if


[
(π
)
]


1

if

2a [1 + (2 − γ cot γ ,

)
]


a sin γ
π
1

cot γ +

2a 1 − 2 + γ − 2 arcsin
y



2
2
2

a
sin
γ+y
cos
γ

if
+ y2 √y2 −a2 sin2 γ ,
[
(
)
]
f (y) =

− 1 1 + π2 − γ cot γ +

 2a
2



+ 2 √ a2sin 2γ 2 ,
if


y
y
−a

[
( sin γ
)
]


a sin γ
1

cot γ +


4a −1 + γ − 2 arcsin
y


2
2
2

a
sin
γ+y
cos
γ
+ √
,
if
2
2
2
2
2y
y −a sin γ
13
y < 0 or y ≥ 2a cos γ2
y ∈ [0, a sin γ)
y ∈ [a sin γ, a)
y ∈ [a, 2a sin γ2 )
y ∈ [2a sin γ2 , 2a cos γ2 ),
(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, Cambridge University Press, Cambridge.
[3] Gille, W. (1988). The chord length distribution of parallelepipeds with their limiting
cases, Exp. Techn. Phys. 36, 197–208.
[4] Aharonyan, N. and Ohanyan, V. (2005). Chord length distribution functions for polygons, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences). 40, no. 4, 43–56.
[5] Harutyunyan, N. and Ohanyan, V. (2009). Chord length distribution function for regular polygon, J. Appl. Prob. 42, no. 2, 358–366.
[6] Stoyan D. and Stoyan,H. (1994). Fractals, Random Shapes and Point Fields, John
Wiley& 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 distribution
of pentagonal and hexagonal rods: relation to small-angle scattering. In Journal of
Applied 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 Sehnenlängen an Ebenen und räumlichen Figuren, 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,
14
Harutyunyan, Ohanyan
Fig. 1. Chord length distribution and density functions for rhombus with a = 1 and γ = π
6.
Chord length distribution function...
15
Fig. 2. Chord length distribution and density functions for rhombus with a = 1 and γ = 5π
12 .