J. Appl, Prob. 6,430-441 (1969)
Printed in Israel
RANDOM PATHS THROUGH CONVEX BODIES
RODNEY COLEMAN, Imperial College, London
Summary
Several mechanisms under which the randomness of straight line paths
through convex bodies can arise are described. Some general results are given
relating four of these mechanisms, and the corresponding distributions of
the lengths of the straight line paths are found for the circle, the rectangle,
and the cube.
1. Introduction
There are many ways in which the randomness of the secants of a convex body
might arise (see, for example, the introduction to Chapter 1 of Kendall and Moran
(1963)). We shall call the convex body K.
(i) Surface radiator randomness (S-randomness). A secant of K is defined
by a point on its surface and a direction. The point and direction are from independent uniform distributions. The symbols Ps( . ), E s are used to denote a density,
expectation respectively under this randomness. This randomness is the main
consideration of Horowitz (1965).
(ii) Interior radiator randomness (I-randomness). A secant is defined by a
point in the interior of K and a direction. 'The point and direction are from independent uniform distributions. 'The density and expectation are denoted by PIC • ),
E I respectively under this randomness.
(iii) Mean free path randomness tp-randomnessi. A secant is defined by its
point of intersection with a fixed plane, and by the direction it makes with that
plane. The point is chosen uniformly over the plane, and the direction independently such that the probability measure is reflection invariant with respect to
the surface of K. (In 2-space the secant is defined by its point of intersection with a
fixed line and its direction.) We bound the measure by considering only the class
of points and directions which define a secant of K. The density and expectation
will be denoted by pp,( . ), Ep, respectively. This randomness was investigated by
Kingman (1965) in connection with the mean free path of a particle projected
from the interior of a convex body K with perfect reflection from the inner
walls.
Received in revised form ,16 December 1968.
430
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431
Random paths through convex bodies
(iv) A secant is defined as the straight line through two points chosen uniformly
and independently in the interior of K. Some results are given by Santal6 ((1953,
page 16) and by Matern ((1960), page 24).
(v) If the boundary of K has nonzero curvature almost everywhere, then a
random secant may be defined as the straight line between two points chosen
uniformly and independently on the surface of K. This randomness has been
considered by Matern ((1960), page 122). It will not be considered further in this
paper.
2. Some general results
Result (i). These secants of a convex body K are Jl-random if when extended
they are the S-random chords of a sphere (in 2-space: circle) K' of very large
radius having K near its centre.
Proof.
(See Figure 1).
Figure 1
We shall consider just the 2-space case: circle K', centre 0, of fixed large radius r,
We take a diameter L to be a fixed line. Under the S-randomness of chords of K', a
chord C is defined by an end P uniform round K', and by the independently uniform
angle t/J which C makes with the radius vector OPe We let 4J, the angle LOP,
define point P. Then
(1)
ps(4J, t/J) oc 1 (0 < 4J < 2n, - in < t/J < tn).
We suppose that C cuts OL at point M. We transform density (1) to the joint
density of x and f), where x is OM and f) is angle PML. We find that
sin f)
p(x, 0) oc ( I_X 2 SIn
• 2f))1/2 ;
r2
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432
RODNEY COLEMAN
and for very large r
p(X,O) = Pjl(x, 0) oc sin 0,
the Jl-randomness density.
Result (ii). The mean length of Il-random secants is 4VjS, where V is the
volume and S the surface area of the convex body K. (In 2-space the analogous
result is EJll = nAjP, where A is the area of K, and P is the length of its perimeter.)
This result is derived in Kingman (1965) only implicitly, and so is here demonstrated explicitly in the 2-space case.
Proof. We consider a fixed line L passing through a fixed point 0 in the interior
of the convex body K, and a line L o through 0 making angle 0 with L, where
-tn < 0 ~ tn. We consider the class Lo of all secants of K which are perpendicular to L o. Any secant in L o may be assigned a coordinate Y, the coordinate on the
axis L o of its point of intersection with the secant (taking 0 to be the origin of
coordinates). We denote by Y~ and y; the inf and sup respectively of y for the
class I o. By definition, the reflection invariant density of all the secants of K is
PJl(y,O) = C
(-tn < 0 < in, y~ < y <
y;),
\\ here C is a constant given by
1= C
l
t 1t
(y; - y~)dO
o=-t1t
= CP,
and, since y; - y~ is the length of a diameter through 0, P is the length of the
perimeter of K. Then
E) = p-l
r"
Jo= -t1t
i'__
8,
l(y,O)dydO,
y0
= p-1An,
where
is the area of K.
Result (iii). (Kingman). The density PI(I) of secant lengths under I-randomness is proportional to I times the density pjl(l) under Il-randomness, i.e.,
(2)
Proof in 2-space. In the coordinate system used in proving result (ii)
PJl(y,O) o: 1
(-tn < 0 < tn, y~ < Y < y;).
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433
Random paths through convex bodies
Under I-randomness we let the interior point on the random secant have rectangular coordinates (u, v) about 0 with the u-axis along L, and then, if the
secant is normal to L o ,
p/(u, v, ()) oc 1 (-!n < () < tn, (u, v) E K).
We rotate the coordinate axes
u
= x cost) - y sinf',
v
= x sin () + ycos(),
() = (),
so that the y-axis is along L o; then
PI(X,Y,()) o: 1 (-tn < () < tn, y~ < y < Y;, x' < x < x"),
where (x" - x') is the length of the secant. The result follows from integrating out x.
From (2), by determining the proportionality constant, we deduce that
(3)
EJlI = [E 11- 1]-1;
and by convexity, we conclude that, for all convex bodies,
(4)
EJlI ~ Ell,
with equality if and only if 1 is constant with probability one. This is Kingman's
corollary.
Result (iv). In d-space, the density piv(l) of secant lengths under the randomness
described in Section l(iv) is proportional to
1 times the density pp(/) under
p-randomness, i.e.,
r:
(5)
piv(I) oc
r:
1 PJl(I).
The method of proof is that used to establish result (iii). This result was brought
to the notice of the author by the referee. It follows from the results leading to
(4) that, for all convex bodies,
Ell ~ Eivl,
with equality if and only if I is constant with probability one.
3. A method of deriving the distribution of the random secant lengths of a convex body
For any given shape of convex body K we set up a coordinate system within
which the specified randomness of any secant can be described by a density.
Instead of examining the class I of all secants of K we can sometimes restrict
ourselves (by using the symmetries of K) to a subset L' of L having the same
distribution of secant lengths. Within L' we determine a mapping from the set of
coordinates onto the length of the secant defined by these coordinates. We then
transform the randomness density from the space of coordinates to the space
of secant lengths.
We consider the random secants of circles, rectangles and cubes.
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434
4.
RODNEY COLEMAN
The unit circle
The results are brought together in Table 1.
(i) S-randomness. We consider the subset of secants (chords) through a
fixed point P on the perimeter of a disc of unit radius. We let (J ( - tn < (J < !n)
be the direction a secant through P makes with the diameter through P. We
consider only the further subset of secants having 0 < (J < tn. Then
2
(6)
Ps«(J) =
-n
and the secant length is I
= 2 cos o.
(0 < (J <
tn),
(ii) I-randomness. We let the interior point P have plane polar coordinates
r, ¢; and let a secant through P make angle (J( -tn < (J < tn) with the radius
vector to P. Then
(0 < r < 1, -n < ¢ < tt, -tn < (J < tn).
The secant has length I = 2 (1 - r2 sirr' (J)"t, so we need consider only the subset
¢ = 0, 0 < (J < tn. Then
PIer, (J)
(7)
r
= -tt
(0 < r < 1, 0 < (J < tn).
TABLE
1
Random paths through a disc of unit radius
The density of the random path lengths is found to be
PA(l) = CAlf A
A
CA
s
1
tt
(1 - ~2rt
(0 < 1<2).
EAl
fA
0
4
tt
= 1.274
1
4
1
n
2
1.571
I
1
2n
2
16
3n
1.697
iv
1
,6n
4
256
45n
1.811
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435
Random paths through convex bodies
5. The rectangle a x b
The results are brought together in Section 5.1.
(i) S-randomness. We consider a rectangle v x w with a secant having its
random end in a side of length v, at a distance r (0 < r < v) from a fixed corner,
and making angle (} (0 < (} < n) with the side. We consider only the subset of
secants for which 0 < (} < tn; then the randomness density is
2
Ps,v (r, fJ) = -no
(8)
(O<r<v,O<fJ<tn),
and the secant has length
r sec (}
(0 < (}< arctan (wlr)),
1- {
w cosec fJ (arctan (wlr) ~ fJ < tn).
(9)
Choosing an end of a secant uniformly over the perimeter of a rectangle a x b,
we take
a
.
b
(10)
Ps(l) = --b Ps a(l) + -b- Ps b(l)·
a+
'
a+
I
The determination of Ps(l) by a geometrical argument is the main result of Horowitz (1965).
(ii) I-randomness. We take the interior point P to be (u, v) (0 < u < a,
and let the secant through P make angle (} with v = constant. We
consider the subset of secants having 0 < (} < -in and u]a < vjb. The randomness
density is then
o < v < b),
4
p/(u, v, fJ) = --b
na
(11)
(0 < u < a, 0 < v < bu]a, 0< (} < tn).
The secant has length
a sec fJ
(12) 1 =
(a - u) sec (}
b cosec fJ
5.1.
(13)
(0 <
e ~ arctan (vlu)),
+ v cosec (}, (arctan (vju) < (} < arctan [(b
- v)j(a - u)J),
(arctan [(b - v)j(a - u)J ~ (} < tn).
We find that under each randomness, i.e., for A
= S, u, I, iv,
PA(l) = PA(l;a, b) + PA(l; b, a),
where
(14)
(0 < I
(a
~
<I<
a),
(a
2
+ b2 }! ),
(and C A is a constant factor). We may thus write
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436
RODNEY COLEMAN
(15)
where
(16)
We can sometimes choose an h~(a, b) simpler than hA(a, b) such that
h~(a,b) + h~(b,a)
= hA(a,b) + hA(b,a).
The values of CA' fA, gA(/; a, b) and h~(a, b) are given in Table 2.
TABLE
2
Random paths through the rectangle a x b
Writer={a 2+b2)t , S={l2_ a2)-·t and Q=10 g(b:r).
h~(a, b)
A
s
n(a
2(a
I
IV
2
o
1
o
+ b)
+ b)
1
abs
-1-
1
-is
ta[(a
2 na
1
+ 2b)q + a -
rJ
b ( .
E I _
nab )
since Jl - 2(a + b)
1
nab
1
3
3a 2b 2
For the square with unit side we find that
Esl
=
!-[31og{1
tt
+ .../2) -
.../2 + 1]
0.710,
= 0.785,
(17)
4
E]I
= 3n[31og (1 +
Ei)
= "3 [I!lg(1 + .../2) + (2 + .../2)/5]
~2) - ~2
+ 1]
0.946,
and
2
1.043.
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437
Random paths through convex bodies
6.
The unit cube
(i) Ssrandomness. We consider the subset of secants having their random
end (x, y, 0) in face z = 0 of the cube {(x, y, z): 0 < x, y, z < 1}. We let the direction
of a secant be (0, t/J) (0 < 0 < tn, -n < t/J < n) where 0 is the angle between the
secant and z = 0, and t/J is the angle between a fixed line, x = constant, in z = 0
and the projection of the secant on to z = O.We consider the subset of these
secants which would, when projected through z = 1 if necessary, meet y = 0
with positive base angle t/J (i.e., 0 < t/J < arctan (xl y)). The randomness density is
then
4
(18) Ps(x,Y,O,t/J) = -cosO (0 < X,Y < 1,0 < 0 < tn, 0 < t/J < arctan (x/y)).
tt
The secant has length
y sec 0 sec t/J (0 < 0 < arctan (y-lcos t/J)
1= {
cosec 0
(19)
(arctanfyr l cos e) ~ 0 <
tn).
Since [ does not depend on x, and depends on y only through w = y sec t/J, we
integrate x from (18), and after setting
w = y sec t/J,
0 = 0, y = y,
in the resulting density we integrate out y. This leaves
(20)
(n/4) sec () piw, ()
=
r1- tw
1
.1.
2
w
tn, 0 < w ~
(0 < 0 <
_ (w2
-
w
1)t
(0 < 0 <
1)
tn, 1 < w < J2).
We now write
(21)
PS(w,O)
= PS,l(W,O) + PS,2(W,0),
where PS,l(W,O) is PS(W,0) over the (w,O) for which [= wsecO, and PS,2(W,0) is
Ps(w,0) over the (w,O) for which [= cosecO.
·We transform PS.l(W,O) by setting [=wsecO, W=W and integrating out w,
to obtain
3n[2 - 41 3
n
+2-
- 2
6[2
+ 6/ 2 -
(0
+ 8al2 + 4a - 121 2 arctan a
4bl 2
-
4b - 6a 2arctan (
3
< 1 ~ 1)
(1 < I ~ J2)
[2) ( -)2 < 1 < -)3)
2~ -
where a = ([2 - 1)+ and b = ([2 - 2)t.
We integrate PS,2(W,0) w.r.t. w, and transforming the result by setting 1 = cosec (J
we obtain
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438
(23)
RODNEY COLEMAN
2ps
n1
,
2(1) = {
n -1
'It -
+ 12 -
1 - 12
(1 < 1 ~ J2)
4a
+ 4b
(~2 < 1 < ~3).
- 4 arctan b
Then
Ps(l) = PS,1 (I) + PS,2(1)·
(24)
The density Ps(l) is illustrated in Figure 3.
It is easiest to find Esl directly from density (20). Then
(25)
(1
+
Esl = Jw=o(1- tw)g(w)dw
(v'2 (
JW=l tw -
(w2
w
1Y!-)
g(w)dw,
where
4
g(w) = -[warctan(ljw)
n
(26)
+ -!log(1 + w2 ) ] ;
and so, after integrating wherever possible w.r.t. w in (25),
1
Esl = 31t[2n
+ 2.J2 -
(27)
-
= 0.598,
6
7
+ 210g2 + 21og 3 -
f
J2
1
(W2 -
w
4.J2arctan(~2)]
1)t
g(w)dw,
after numerical quadrature of the remaining integral. Horowitz ((1965), page 176)
corrected in Horowitz (1966), gives a formula for Esl which reduces to the expression (27).
(ii) I-randomness.
See Figure 2.
0'
0
Figure 2
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439
Random paths through convex bodies
The secant passes through an interior point (x,Y,z) of the unit cube {(x,y,z):
the y-axis in the x-y (horizontal) plane,
and () to the horizontal in the vertical plane. Then
o< x, y, z < 1}, and makes angles l/J to
(28)
PI(X, y, Z,
f),
(0 < x, y,
l/J) ex: cos f)
Z
< 1, --!n <
f)
< tn, 0 < t/J < 2n).
We consider the projection of the secant on the x - y plane where we take the
subset of secants for which x is greater than y. The situation is that of a secant,
of length wand having its random interior point (x, y) a distance q from one end,
in a unit square under randomness density
4
PI(X, y, t/J) = n
(29)
(0 < x < 1, 0 < Y < x, 0 <
t/J < in).
We find that
1
(0 < w ~ 1, 0 < q < w)
(30)
2
-1)t
(1 < w < .)2,0< q < w),
-2(W
-2- - 1
W
I
and from this we determine PI(W) and PI(q w).
We now consider the vertical plane through the secant. The situation is that
of a secant of length 1 through interior point (q, z) of a rectangle w x 1 under
randomness density
I
I
PI(Z,(J, q w) ex: cos (J PI(q w).
(31)
We find PI(II w) from this, and average w.r.t. PI(W) to obtain
81 3
(32)
2n1 2 PI(1) =
-
314
(0 < 1 ~ 1)
6n + 61 4
-
1 - 8(21 2
6n - 31 4
-
5
+ 1) (1 2 -
+ 8(1 2 + 1) (1 2 -
(1 < 1 ~ .)2)
1)1-
2) 1- - 24 arctan (1 2
-
2)1-
(')2 < 1 < .)3).
From results (iii) and (iv) of Section 2 we have also that
(33)
These distributions are illustrated in Figure 3.
We find that
(34) Ell
= 1 + J.. 1og 2 +
n
(~2 - 4n
~)10g 3 -
4')2 arctan
tt
6
-tt
(~)
-J2
f
1
arctan.Jw
w=o
w+2
dw
0.896;
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440
RODNEY COLEMAN
i
(35) E,)
(directly from result (ii) of Section 2);
and
4
tt
17
2
-63 - -9 + -63 y /2 - 7-
(36) Eivl
y
/3
1
2
+ -log(l
+ y /2) + -log(2
+
3
3
y
/3)
1.103.
ptt)
24~
2531 3796 3 ·975
22j
2-0
I
16-
1-4
1·2
1-0
08
0·6
0·4
0'2
Path Length
,
Figure 3
The densities of random paths through a cube with unit side
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Random paths through convex bodies
7.
441
Acknowledgements
I wish to thank my research supervisor, Mr. A. D. McLaren, for his encouragement and advice. Most of this work was done while the author held a Science.
Research Council studentship at Churchill College, Cambridge. The University
of Cambridge Mathematical Laboratory's TITAN computer was used to determine
most of the numerical results. I thank the referee for bringing to my notice the
result (iv) of Section 2, the use of which has considerably exended the generality
of this paper.
References
HOROWI1Z, M. (1965) Probability of random paths across elementary geometrical shapes.
J. Appl. Probe 2, 169-177.
HOROWITZ, M. (1966) Correction to HOROWITZ (1965). J. Appl. Probe 3, Opp, 284.
KENDALL, M. G. AND MORAN, P. A. P. (1963) Geometrical Probability. London, Griffin.
KINGMAN, J. F. C. (1965) Mean free paths in a convex reflecting region. J. Appl, Probe 2,
162-168.
MATERN, B. (1960) Spatial variation. Medd. fran statens skogsforskningsinstitut 49: 5.
SANTAL6, L. A. (1953) Introduction to Integral Geometry. Paris, Hermann.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 18:33:24, subject to the Cambridge Core terms of use,
available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0021900200032939