Acc. Sc. Torino – Atti Sc. Fis.
140 (2006), 83-90.
GEOMETRIA
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Riassunto. Si risolvono problemi di tipo Buffon per un corpo test
arbitrario e una speciale configurazione di linee nel piano Euclideo.
Abstract. We solve, in the Euclidean space E2 , a problem of Buffontype for an arbitrary “test body” K and a lattice of lines whose elj
ementary tile C0 = ∪tj=1 S (j) , where S (j) = ∪m
i=1 Pi is a strip of m
parallelogram Pij of sides aj and bi (i = 1, ..., m) and acute angle
αj ∈]0, π/2].
1. Introduction
In [1] an arbitrary fixed convex set in E2 is considered as are two
families of equally spaced parallel lines making angle α with each
other. It is assumed that the inter-line distance in each family of
parallel lines is greater than the maximum width of the convex set.
A congruent copy of the convex set is placed randomly.
A simple formula for the probability that the randomly placed
set intersects at least one of the lines is obtained. A consequence
of the formula is that there exists at least one angle α (depending
on the convex set) such that the event of intersecting some line in
one of the two families of parallel lines is independent of the event of
intersecting some line in the other family. In this paper we consider
a generalization of the Buffon needle problem for an arbitrary convex
test body K and a lattice of lines whose fundamental cell is a set
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Giuseppe CARISTI - Giovanni MOLICA BISCI - Alfio PUGLISI
GIUSEPPE CARISTI, GIOVANNI MOLICA BISCI E ALFIO PUGLISI
of parallelograms. We give an exact expression for the intersection
probability of a randomly tossed test body K with the planar lattice.
Moreover, some independence questions are studied.
More precisely let K be a “test body” and R be a lattice of lines
j
whose elementary tile C0 = ∪tj=1 S (j) , where S (j) = ∪m
i=1 Pi is a strip
of m parallelogram Pij of sides aj and bi (i = 1, ..., m) and acute angle
αj ∈]0, π/2].
When K is tangent to an oriented line g, then Sg will denote the
orthogonal projection of S on g, and if ϕ is the angle between a given
direction d related to the body and SSg , we set p(ϕ) := |SSg |, the
distance from S to g.
The 2π-periodic extension function p : R → R will be called the
support function with respect to the pair (K, d). We denote by L the
function L : R → R given by L(ϕ) := p(ϕ) + p(ϕ + π). We call L
the width of the pair (K, d) in the direction ϕ. By construction L is
a π-periodic function.
We denote by M the set of all convex test bodies congruent to K
and with barycenter S within C0 . We also assume that these convex
test bodies are uniformly distributed, i.e. that the coordinates of S
are a bidimensional random variable with uniform distribution in C0 ,
and that the random variable ϕ is uniformly distributed in [0, 2π], S
and ϕ stochastically independent.
Finally we denote by N the set of convex bodies K, of diameter
Diam(K), which are completely contained in C0 . As is well known
then we can write the probability that the test body K intersects the
boundary of one of the tiles of the lattice R:
(1)
pK = 1 −
µ(N )
,
µ(M)
where µ is the Lebesgue measure.
The measures µ(N ) and µ(M) can be computed using the elementary
Kinematic measure in E2 [[5],p.126]
(2)
dK = dx ∧ dy ∧ dϕ,
Random convex bodies and parallelograms strips
RANDOM CONVEX BODIES AND PARALLELOGRAMS STRIPS
85
where x and y are the coordinates of P ∈ K and ϕ is an angle of
rotation.
2. Main results
Now consider for fixed ϕ ∈ [0, π] the set of points P ∈ C0 for with
the body K with centroid P does not intersect the boundary ∂C0
and let C(ϕ) the topological closure of this open subset of C0 . In the
sequel we will assume that the body K is small 2 with respect to the
lattice R, using some restriction on the diameter Diam(K) of K:
Diam(K) < min aj sin αj , bi sin αj : i ∈ {1, ..., m}, j ∈ {1, ..., t} .
Theorem 1. The probability that a convex body K of boundary of
length L, intersects one of the lines of the lattice R is
m
1
m
L m
pK = bi + t
bi +
t
m
t
π
a
b
sin
α
j
i
j
i=1
i=1
j=1
i=1
j=1
(3)
π
t
1
L(ϕ)L(ϕ + αj )dϕ .
−m
sin
α
0
j
j=1
Proof. Let us consider the fundamental cell C of the lattice R.
(j)
We denote by Ni the set of all “test bodies” K whose barycenter
are inside in the parallelogram Pij .
Thus
pK = 1 −
(4)
t
j=1
(j)
i=1 µ(Ni )
m
µ(M)
,
where
µ(M) = 2π
t
j=1
2
aj
m
t
bi
i=1
j=1
sin αj .
We say that the body K is small with respect to R, if the polygons sides of
C(ϕ) and C0 are pairwise parallel.
Giuseppe CARISTI - Giovanni MOLICA BISCI - Alfio PUGLISI
GIUSEPPE CARISTI, GIOVANNI MOLICA BISCI E ALFIO PUGLISI
86
(i,j)
Let Rϕ
lengths
be the rectangle with sides parallel to those of Pij , of
aj − L(ϕ)/ sin αj , bi − L(ϕ + αj )/ sin αj .
We can write
(j)
µ(Ni )
=
2π
0
dϕ
(i,j)
dxdy =
(x,y)∈Rϕ
2π
[aj − L(ϕ)][bi − L(ϕ + αj )]
sin αj
0
dϕ
By the fact that L is a π-periodic function and by Cauchy formula,
i.e.
2π
L(ϕ)dϕ = 2L,
0
we get
m
t j=1 i=1
(j)
µ(Ni )
= 2π
t
aj
j=1
m
t
bi
i=1
t
1
+2m
sin
αj
j=1
π
0
sin αj −2L m
j=1
t
aj +t
j=1
m
i=1
L(ϕ)L(ϕ + αj )dϕ .
When we replace the expression
the probability (3).
t
j=1
(j)
i=1 µ(Ni )
m
in (4) we have
Corollary 2. If S is a segment of constant length, the probability
that S intersects one of the lines of the lattice R is
(5)
1
t
2l
pK = m
t
t
π
i=1 bi
j=1 sin αj
j=1 aj
j=1
aj +
m
i=1
bi +
bi +
Random convex bodies and parallelograms strips
RANDOM CONVEX BODIES AND PARALLELOGRAMS STRIPS
−ml2
t 1+
j=1
π
− αj cot αj
2
87
.
If E is an ellipse of half-axes ξ and ζ, the width function is given
by
L(ϕ) = 2 ξ 2 sin2 ϕ + ζ 2 cos2 ϕ.
Hence formula (3) gives the following
Corollary 3. The probability that a random ellipse E of boundary
of length L, intersects one of the lines of the lattice R is
t
1
m
L m
aj + t
bi +
pE = t
m
t
π
j=1
i=1
j=1 aj
i=1 bi
j=1 sin αj
(6)
t
1
×
−4m
sin αj
j=1
π
(ξ 2 sin2 ϕ
+
ζ 2 cos2 ϕ)(ξ 2 sin2 (ϕ
0
+ αj ) +
ζ 2 cos2 (ϕ
+ αj ))dϕ
.
The expression (6) extends the formulas proved in [6] and in [4].
A particular case of the previous result is the following
Corollary 4. If Σ is a disk of constant diameter D, the probability
that Σ intersects one of the lines of the lattice R is
(7)
pΣ,R = D m
t
t
j=1 aj
m j=1 aj
+t
t
j=1 aj
i=1 bi
t
j=1 sin αj
D2
m
m
i=1 bi
t
m
i=1 bi
1
.
sin2 αj
j=1
−
Giuseppe CARISTI - Giovanni MOLICA BISCI - Alfio PUGLISI
GIUSEPPE CARISTI, GIOVANNI MOLICA BISCI E ALFIO PUGLISI
88
In general for a convex body of constant width k, using Cauchy
relation, we have
L=
1
2
2π
kdϕ = πk.
0
By this fact we obtain
Corollary 5. If K has constant width k, the probability that K
intersects one of the lines of the lattice R is
(8)
pK,R = m
t
m
aj + t
j=1
i=1
− k
bi t
j=1 aj
m
t
j=1 aj
m
i=1 bi
m
i=1 bi
t
t
j=1 sin αj
k2
.
sin2 αj
j=1
+
3. Dependence structure of the hitting events
In this section, considering α ∈]0, π/2], we assume αj = α, ∀j ∈
{1, ..., t}.
We can look at the lattice R as the superposition of two elementary lattices of parallel lines: the lattice Ra of lines parallel to the side
b1 of P1 , with distances ai sin α, and the lattice Rb of lines parallel
to the side a1 of P1 and with distances bi sin α. Hence
R = Ra ∪ Rb .
We denote by Ea the event “a body test K intersects one of the lines
of Ra ” and by Eb the event “a body test K intersects one of the lines
of Rb ”.
Theorem 6. The probability p∗K,R that a convex body K intersects
at the same time two lines with different directions in the lattice R
is
Random convex bodies and parallelograms strips
RANDOM CONVEX BODIES AND PARALLELOGRAMS STRIPS
p∗K,R
(9)
mt
= t
m
2
π
j=1 aj
i=1 bi sin α
π
89
L(ϕ)L(ϕ + α)dϕ.
0
Proof. By simple remarks, using the same arguments in [2] and
denoting by p(Ea ) and p(Eb ) the probabilities of the events Ea and
Eb , we have
tL
p(Ea ) = ,
t
b
π
j=1 i sin α
But
1
mL
p(Eb ) = .
m
b
π
i=1 i sin α
t
m
p(Ea ∪ Eb ) = L m
aj + t
bi +
m
t
a
b
sin
α
π
j
i
j=1
i=1
i=1
j=1
mt
−
sin α
π
L(ϕ)L(ϕ + α)dϕ .
0
Hence the probability p(Ea ∩ Eb ) that K meets at the same time
some line in Ra and some line in Rb is
p(Ea ∩ Eb ) = p(Ea ) + p(Eb ) − p(Ea ∪ Eb ).
Substituting the previous formulas we get the assertion.
Finally, imposing the condition of independence for the events Ea
and Eb we get
Theorem 7. The events Ea and Eb are independent if and only if
(10)
π
0
L(ϕ)L(ϕ + α)dϕ =
L2
.
πmt
90
Giuseppe CARISTI - Giovanni MOLICA BISCI - Alfio PUGLISI
GIUSEPPE CARISTI, GIOVANNI MOLICA BISCI E ALFIO PUGLISI
Hence immediately we have
Corollary 8. If Σ is a circle of constant radius δ with
sin α
min{a1 , ..., at , b1 , ..., bm },
2
the events Ea and Eb are independent if and only if (m, t) = (4, 4).
δ<
Finally
Corollary 9. If K has constant width k, the events Ea and Eb are
independent if and only the fundamental cell of the lattice R is given
by a single parallelogram of sides a and b and acute angle α ∈]0, π/2].
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et
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,
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Giuseppe Caristi and Alfio Puglisi
Department of Economic and Business Branches of Knowledge.
Faculty of Economics, University of Messina,
98122 - (Me) Italy.
E-mail: [email protected]