Random system of lines in the Euclidean plane E2
Giuseppe Caristi
Abstract. In this paper we consider a random variable arising from a
problem of geometrical intersection between a fixed convex body K and
a system of random lines in E2 .
M.S.C. 2000: 60D05,52A22.
Key words: Geometric probability, stochastic geometry, random sets, random convex
sets, integral geometry.
1
Introduction
Let E2 be the Euclidean plane and let K be a convex non empty and bounded domain
of area SK and with boundary ∂K of length L. We consider a family F of random,
uniformly distributed n-lines {G1 , ..., Gn } with n ≥ 2. We assume that if Gh , Gk ∈ F ,
f . It is possible that this points belongs to K or not. In this way
then Gh ∩ Gk 6= ¡
we have a random variable X(n,K) . In this paper we give the following result
³
´
¡
¢
k
Theorem 1. The expression of the mean value E X(n,K) , the k-moments E X(n,K)
¡
¢
and the variance σ 2 X(n,K) of the random variable X(n,K) can be calculated as follows
#
"
³
´
X
¡
¢
SK
πSK
k!
k
E X(n,K) = απ 2 ,
E X(n,K)
=
,
L
J1 !...Jα ! L2
J1 +...+Jα =k
¡
¢ πSK
σ 2 X(n,K) =
L2
µ
¶
πSK
1− 2
α2 ,
L
where
α=
n (n − 1)
,
2
and k is a positive integer.
Other results about the computation of the variance are investigated in [2] and an
extension in the 3-dimensional Euclidean space of the same problem is studied in [6].
Applied Sciences, Vol.10, 2008, pp.
48-53.
c Balkan Society of Geometers, Geometry Balkan Press 2008.
°
Random system of lines
2
49
Main Results
Let N be the set of natural numbers, n ≥ 2 a fixed natural integer, {G1 , ..., Gn } and
K as in introduction. We can state the following
Theorem 2. Let us consider the random variable X(n,K) . Then
¢
¡
SK
E X(n,K) = απ 2 ,
L
where α =
K.
n(n−1)
,L
2
is the length of ∂K and SK is the area of the domain defined by
Proof. It is easy to see that, denoting with L the length of ∂K and dG the elementary
measure of the lines in the Euclidean plane E2 , we have
Z
½
¾ dG = L.
f
G∩K6= ¡
Since G1 , ..., Gn are stochastically independent, we get
Z
¾ dG1 ∧ ... ∧ dGn = Ln .
½
f
G∩K6= ¡
If we consider the lines G1 , ..., Gn , then the intersection points might belong to K or
not. Hence we obtain a random variable which we denote by X(n,K) .
In order to compute the variance, we have the following integral
Z
¾ X(n,K) dG1 ∧ ... ∧ dGn
½
f
G∩K6= ¡
We define the application ²hk = 1 if Gh ∩ Gk ∈ K (with h 6= k = 1, ..., n) and zero
otherwise. Then
X
X(n,K) =
²hk .
h,k=1
Further, let us consider
Z
I2 :=
½
f
Gh ,Gk ∩K6= ¡
¾ ²hk dGh
∧ dGk .
If (Gh ∩ Ek ) ∈ K, then we have ²hk = 1. We denote with λk the chord intercepted
by Gk on K (and its length).
We have


Z
Z
Z
½
¾ ² dG ∧ dGk = ½
¾ dGh  dGk ,
¾ ½
f hk h
f
f
Gh ,Gk ∩K6= ¡
Gk ∩K6= ¡
Gk ∩K6= ¡
but it is well known that
Z
½
f
Gh ∩λk 6= ¡
¾
dGh = λk ,
50
Giuseppe Caristi
Z
½
f
Gk ∩K6= ¡
¾
Moreover,
Z
½
f
G∩K6= ¡
λk dGk = πSK .
Z
¾
X(n,K) dG1 ∧ ... ∧ dGn =
X
½
f
G∩K6= ¡
¾
²hk dG1 ∧ ... ∧ dGn =
h,k=1
πSK Ln−2 + ... + πSK Ln−2 .
Taking into account that the number of the different sets {Gh , Gk } is (the binomial
coefficient)
n (n − 1)
α=
,
2
we have
Z
½
¾ X(n,K) dG1 ∧ ... ∧ dGn = απSK Ln−2 .
f
G∩K6= ¡
By definition,
R½
¡
¢
E X(n,K) :=
f
G∩K6= ¡
R½
¾
X(n,K) dG1 ∧ ... ∧ dGn
f
G∩K6= ¡
¾
dG1 ∧ ... ∧ dGn
,
³
´
2
and hence mathbf E X(n,K)
= απ SLK2 .
Now we compute
³
´
R½
2
E X(n,K)
=
We put
f
G∩K6= ¡
R½
¾
2
X(n,K)
dG1 ∧ ... ∧ dGn
f
G∩K6= ¡
¾
dG1 ∧ ... ∧ dGn
.
Z
J=
½
f
G∩K6= ¡
¾
We can prove that
2
X(n,K)
dG1 ∧ ... ∧ dGn

2
X(n,K)
=
X
2
²hk  ,
h,k=1
and then
2
X(n,K)
=
X
²2hk + 2
h,k=1
X
²hk ²sn .
(h,k)6=(s,n)
With this observations we can compute the integral


Z
X
X
¾
²2hk + 2
²hk ²sn  dG1 ∧ ... ∧ dGn .
J= ½
f
G∩K6= ¡
h,k=1
(h,k)6=(s,n)
Random system of lines
51
Considering that
Z
Z
½
f
G∩K6= ¡
¾ ²2
hk dGh
∧ dGk =
½
f
Gk ∩K6= ¡

Z
¾ ½

f
Gh ∩λk 6= ¡
¾
dGh  dGk ,
where λk is the chord intercepted by Gk on K, we obtain
J = πSK Ln−2 + ... + πSK Ln−2 + 2πSK Ln−2 + ... + 2πSK Ln−2 .
Taking into account that the number of different sets {Gh , Gk } is α and that G1 , ..., G2
are independent, we infer
³
´ πS αLn−2
πSK α2
K
2
E X(n,K)
=
=
.
n
L
L2
We obtain
Theorem 3. Let us consider the random variable X(n,K) . Hence
¡
¢ πSK
σ 2 X(n,K) =
L2
where α =
K.
n(n−1)
,L
2
µ
¶
πSK
1− 2
α2 ,
L
is the length of ∂K and SK is the area of the domain defined by
Moreover, we note that

k
X(n,K)
=
X
k
²hk  ,
h,k=1
implies
³
´
"
k
E X(n,K)
=
X
J1 +...+Jα =k
3
#
k!
πSK
.
J1 !...Jα ! L2
Applications
1. As first case we consider in the plane a square Q of side a. We can compute the
following values
¡
¢ απ
E X(n,Q) =
≈ 0, 196345α.
16
"
#
³
´
X
k!
π
k
E X(n,Q) =
.
J1 !...Jα ! 16
J1 +...+Jα =k
and the variance is
σ 2 X(n,Q) =
π´ 2
π ³
1−
α ≈ 0, 15779α2 .
16
16
52
Giuseppe Caristi
2. Taking in plane a rectangle R of sides a and b we have
¢
¡
E X(n,R) =
³
E
k
X(n,R)
´
"
απab
2,
4 (a + b)
#
k!
π
=
,
J1 !...Jα ! 4 (a + b)2
J1 +...+Jα =k
X
and the variance
σ X(n,Q) =
Ã
π
2
2
4 (a + b)
1−
!
π
4 (a + b)
2
α2 .
3. Let C be a circle of radius δ. We have
³
´ α
k
E X(n,C)
= ,
4
#
"
³
´
X
k!
α
k
E X(n,C)
=
J1 !...Jα ! 4
J1 +...+Jα =k
and the variance is
¡
¢ π³
α´ 2
σ 2 X(n,C) =
1−
α ≈ 0, 16855α2 .
4
4
4. As last case we consider an equilateral triangle T of side a, obtaining,
³
´ απ √3
k
≈ 0, 3023α,
E X(n,T ) =
18
"
# √
³
´
X
k!
π 3
k
E X(n,T ) =
,
J1 !...Jα ! 18
J1 +...+Jα =k
and for the variance the expression
σ
¡
2
X(n,T )
¢
√ Ã
√ !
π 3
π 3
=
1−
α2 ≈ 0, 2109α2 .
18
18
Remark 4. We observe that in examples 1, 3, 4 the functions are independent of the
dimension of the convex body.
References
[1] W. Blaschke, Vorlesungen über Integralgeometrie, III Auflage, V.E.B. Deutscher
Verlag der Wiss., Berlin 1955.
[2] G. Caristi amd G. Moltica Bisci, On the variance associated to a family of ovaloids
in the Euclidean Sapce E3 , Bollettino U.M.I. (8) 10-B (2007), 87-98.
Random system of lines
53
[3] E. Czuber, Zur Theorie der geometrischen Wahrscheinlichketein, Sitz. Akad. Wiss.
Wien 90 (1884), 719-742.
[4] R. Deltheil, Probabilitiés géométriques, Gauthier-Villars, Paris 1926.
[5] A. Duma and M. Stoka, Probabilità geometriche per sistemi di piani nello spazio
euclideo E3 , Rend. Sem. Mat. Messina 7 (2000), 5-11.
[6] G. Molica Bisci, Random systems of planes in the Euclidean space E3 , Atti Acc.
Scienze di Torino 2005, 43-48.
[7] H. Poincaré, Calcul des probabilitiés, Ed. 2, Carré, Paris, 1912.
Author’s address:
Giuseppe Caristi
University of Messina,
75 Via dei Verdi, 98122 Messina, Italy.
E-mail: gcaristi@unime.it