random sections of a sphere

The Canadian Journal of Sfatistics Vol. 17, No. 1, 1989. Pages 27-39 La Revue Conadienne de Sfafistique

27

Random sections of a sphere* Rodney COLEMAN

Imperial College, London

Key words and phrases: Spheres, stereology, uniformly random sections, Bertrand para-

AMS 1985 subject classifications: Primary 60D. dox, Wicksell corpuscle problem.

ABSTRACT The Bertrand paradox is that, whereas we can define in a unique way a point uniformly at random

in the interior of a circle, uniformly random chords can be given a variety of competing specifica- tions. This is generalized to spheres, and the distributions of the uniformly random line sections (chords) and plane sections (disks) are tabulated. This includes the large class which are constructed as uniformly random chords of uniformly random disk sections.

RESUME La paradoxe de Bertrand dit que, m&me si on dtfinit de fafon unique la notion de choix altatoire

d’un point il l’inttrieur d’un cercle, on peut dtcrire de plusieurs fasons le concept de cordes uni- fonnbment albatoires. On gtnkralise ceci aux sphkres et les distributions des sections de droites (cordes) et des sections de plans (disques), uniformtment aleatoires, sont tabultes.

1. INTRODUCTION

Bertrand (1888, pp.4,5) showed several distinct ways of defining a chord drawn uniformly at random across a circle. The chord lengths have different probability distributions in each case. When we consider uniformly random sections of a sphere, we have an even wider choice: the uniformly random line sections (chords) and the uniformly random plane sections (disks). We include those line sections that are chords drawn uniformly at random across uniformly random disk sections. We therefore also list the distributions of uniformly random chords of circles, adding to those dealt with by Bertrand.

The persistent and repetitious use of uniformly random, uniformly at random, uniform randomness, and other variants indicates that an abbreviation is required. We shall use UR as the abbreviation in each case.

This paper is a guide to the UR distributions, putting them together in a single collec- tion. Some of the individual results, particularly for the chords of circles, are known, but in practice it would be easier to verify the formulae by deriving them directly rather than by scouring the literature, since each is the result of a simple exercise in geometrical probability. The results for chords in disk sections are all new if we exclude the two general formulae of Davy and Miles (1977). Nevertheless, acknowledgement is due to Kendall and Moran (1963), Kingman (1969), Miles (197 l), Solomon (1978), and Cole- man ( 1979).

*Supported by the University of Calgary (through a special grant from its General Endowment Fund) and the Natural Sciences and Engineering Research Council of Canada (through a grant to Dr. E.G. Enns of the University of Calgary).

28 COLEMAN Vol. 17, No. 1

TABLE 1 : Types of uniform randomness Rx for chords of circles

Rx Description

Surface radiator randomness. One end of the chord is UR on the circumference, and the chord is independently at a UR angle. Bertrand’s first solution.

Invariant randomness. Isotropic uniform randomness. Mean-free-path randomness. The chord is UR from a beam of parallel rays that hit the circle, the beam having a UR orientation. Bertrand’s second solution.

Perpendicular point randomness. The midpoint of the chord is UR in the interior of the circle. Bertrand’s third solution.

Interior radiator randomness. Weighted invariant randomness. A point is chosen UR in the interior of the circle. The chord passes through the point in an independently UR direction.

The chord is through two points independently UR over the interior of the circle.

The chord is through two points independently UR on the circumference of the circle.

The chord is through two points, one UR in the interior of the circle, the other independently UR on the circumference.

A chord is taken with UR Rz through a sphere. The sphere is projected orthogonally onto a plane having a UR orientation. Its projection is a circle. The projection of the chord is a line segment. This is extended to form a chord of the circle. In particular Ta, Tp, Th.

To see the large number of uniform randomnesses possible we have only to examine the lists given in Tables 1, 3, and 5 , and it is not claimed that these exhaust every possible case. In these tables the labels used to identify the randomnesses offer alternative nota- tions. A consistent authoritative labelling is not yet in use, and the alternatives will assist in relating published results one with another.

This paper will be devoted to identifying the sampling procedure implied by the phrase random section and the distributions induced thereby when it is used in applied stereology. Stereology is the measurement of three-dimensional structures from random sections, where interpretation of the data will obviously depend on the precise identification of the type of randomness. Streit (1978) uses the Neyman-Pearson lemma to choose between pairs of distributions in Bertrand’s three cases of uniformly random chords of a circle. This can be extended to cover the complete set of distributions, including plane and line sections of spheres.

In estimating the sphere size distribution in a corpuscule problem (Wicksell 1925), a population of spheres is sampled by a plane section. Here the size distribution of the spheres is related by an integral equation to the size distribution of the circles made in the plane by the spheres that are hit, and in tackling this problem (cf. Coleman 1987a), a general result of Davy and Miles (1977) appeared to be contradicted. By examining several examples given in this paper, it was shown that scale and edge effects were operating, and the problem was resolved by demonstrating a particular invariance property.

A further application is to a problem in mineral liberation: identifying the core size distribution of a population of spherical particles containing cores from data which consist of a single uniformly random section through each of a random sample of particles (Coleman 1987b).

2. UNIFORMLY RANDOM CHORDS OF CIRCLES

The length L of a chord of a circle of unit radius is uniquely determined by the distance X from the centre of the circle to the midpoint of the chord:

1989 RANDOM SECTIONS OF A SPHERE 29

TABLE 2 : Distributions of chord length L for uniformly random chords of circles of unit radius

I 1 4 x - l l

71 - 0.446 2

TA

I

256 6rr 45rr

0.233 - 1 - x - ' 1 4

1024 20rr 175rr x-116 - 0.176

1 - 1 - 1 2

4 3 - 0.471

L = 2 v F F .

The probability density functions therefore satisfy the relationships

The results for the uniform randomnesses of Table 1 are given in Table 2. In the table

We note that, for a circle, the mechanisms S and p are equivalent, as are W, a, and

From (2) the probability density functions for X all take the form

x = V i T p .

Tp , and T and T, also.

p x ( x ) = cxj( 1 - X * ) + k (0 < x < 1) (3)

with j = 0 or 1, i.e., X 2 has a Beta distribution with t ( j + 1) and h(k + 2) degrees of freedom, and so

)L2 = 1 - x2 also has a Beta distribution, with 4 ( k + 2) and $( j + 1) degrees of freedom,

Appendix.

(1978), Ehlers and Enns (1 98 11, and Jaynes ( 1 973).

A common method of derivation for the URs j3, a, T, T,, Tp, and TA is outlined in the

Additional references for this section are Garwood and Holroyd (1966), Enns and Ehlers

3. UNIFORMLY RANDOM PLANE SECTIONS OF SPHERES

The area A of the disk intercept of a plane section of a sphere of unit radius is related to

(4)

the distance Y from the centre of the sphere to the centre of the disk by

A = ~ ( l - Y').


TABLE 3: Types of uniform randomness RY for disk sections of spheres

Rr Description

Invariant randomness. Isotropic uniform randomness. The plane section is UR from a beam of parallel plane sections hitting the sphere, the normal to the beam having a UR orientation.

Interior radiator randomness, area-weighted IUR, area-weighted invariant randomness. A point is chosen UR in the interior of the sphere. A chord passes through the point in a UR direction. The section passes through the point normally to the chord.

Perimeter-weighted invariant random. There is no naturaI mechanism. The probability density function for the perimeter length is pB(b) a bp,(b).

Perpendicular point randomness. The centre of the disk section is UR in the interior of the sphere.

The disk section is through three points independently UR in the interior of the sphere.

The disk section is through three points independently UR, two in the interior and one on the surface of the sphere.

The disk section is through three points independently UR, one in the interior and two on the surface of the sphere.

The disk section is through three points independently UR on the surface of the sphere.

A chord is taken with UR RZ through the sphere. The plane section is taken through this chord normally to the ray from the centre of the sphere to the midpoint of the chord.

A chord is taken with UR Rz through the sphere. The plane section is taken through this chord with a UR orientation.

The plane section is taken through a pair of independent UR chords having URs R,, and Rz2.

The probability density functions are therefore related by

Table 3 lists the types of uniform randomnesses that are applicable to disk sections of spheres, while Tabie 4 lists the distributions of section area A for the various randomnesses.

For the randomnesses I IR~ , the probability density functions for Y are the same as those for the chords, i.e., p y ( y ) = p z ( y ) , where p z ( y ) are given in Sections 4 and 5 which follow.

In Table 4

For each randomness in Table 4 the probability density function for A / T is that of a Beta distribution. The distributions for the p, p a , ap, and Trandomnesses are taken from Miles ( 1 97 1).

4. UNIFORMLY RANDOM LINE SECTIONS OF SPHERES For a straight-line section of a sphere of unit radius, the distance Z from the centre of

the sphere to the midpoint of the chord is related to the length L of the chord by the same equations (1) and (2) as for the chord of a circle. The types of randomness are given in Table 5 .

The distributions of chord length L for the randomnesses of Table 5 are given in Table 6, except for the ! 2 R x , R y randomnesses, which are dealt with separately in Table 7.


TABLE 4: Distribution of the area A of the disk section for uniformly random plane sections of spheres of unit

radius

1 2 n

2

- y - l

3 4n

15 16n

35 32%

- y - ' t 2

- y - ' t 4

- y-116

- 315 y - I t 8

256%

3 - Y 2n

1 - 1-1

2%

1 - 71

3 2n

2 - t*

3 - t4

- 1

71

n

0.937

0.785

0.672

0.518

0.421

0.354

0.823

0.937

0.907

0.823

0.740

0.608

The URs I and f3 for the sphere lead to the same distribution of L (Solomon 1978, Coleman 1981). We note that 4L2 has a Beta distribution exactly as seen for the chord of a circle in Table 2 and for A/T in Table 4.

5. A UNIFORMLY RANDOM CHORD IN A UNIFORMLY RANDOM DISK SECTION OF A SPHERE

We take the sphere to have unit radius. The distribution of the distance Y from the centre of the sphere to the centre of the disk section is given by (5 ) with a probability density function from Table 4. If Y takes value y , the disk section has radius r = The distribution of the distance X from the centre of the disk to the midpoint of a UR chord is

1 P ( X l Y ) = 7 PX (3) (0 < x < t ) , (6)

where pX(x> is given by (2) with a probability density function from Table 2. The joint probability density function for X and Y is therefore


TABLE 5 : Types of uniform randomness Rz for chords of spheres

RZ Description ~~ ~~

s, Y Surface radiator randomness. The chord is taken in a UR direction from a point which is independently UR on the surface of the sphere.

Invariant randomness. Isotropic uniform randomness. The chord is UR from a beam of parallel rays that hit the sphere, the beam having an independently UR orientation.

Interior radiator randomness. Weighted invariant randomness. A point is chosen UR in the interior of the sphere. The chord passes through the point in an independently UR direction.

The chord is through two points independently UR over the interior of the sphere.

The chord is through two points independently UR on the surface of the sphere.

r , CL, IUR

W, Y, WIUR

T ,

P a The chord is through two points independently UR, one in the interior and the other on the

surface of the sphere.

n R V

~ R ~ , R ~

The chord is a diameter of a disk section. The disk section is UR with randomness Ry, and the chord has an independent UR direction in the disk.

The chord of the sphere is a chord of a disk section. The disk section is UR with randomness Ry. The chord has the UR Rx across the disk section.

TABLE 6 : Distribution of the length L for uniformly randomchords of spheres of unit radius

S 1 1 0.577 1, P tl 4 0.471 W 412 t 0.387 a 413 ! 0.327 T &15 4 0.247

n, 3lVG-P 4Tr 0.461

The distance Z from the centre of the sphere to the midpoint of the chord is m. The Jacobian of the transformation ( X , Y ) + ( Z , Y ) is z ( z 2 - Y’ ) -~ , so the joint probability density function for Z and Y is

The marginal probability density function for Z is then

P A 4 = l2 P(Z7Y) dY. 0

As an illustration, when Rx = I and RY = I ,

= z K ( z ) ,

(9)

where K ( z ) is the complete elliptic integral of the first kind.

1989

From (2)

RANDOM SECTIONS OF A SPHERE 33

Table 7 gives the results of taking each Rx of Table 2 with each RY of Table 4. For spheres the symmetry sometimes leads, as we have already seen, to the chord length distribution being the same as that which would result from a quite different uniform randomness mechanism. In Table 7 some of these relationships are given in the column headed R,, which refers to the URs of Table 6 or those elsewhere in Table 7. In Table 7

In = log,, u = ~ 3 - 7 , l + z l + z

1 - z r = In ( T) = 4 In (-) = tanh-' z,

E = E ( z ) ,

the complete elliptic integral of the second kind, and

K = K ( z ) ,

the complete elliptic integral of the first kind.

a sphere: We note in particular the following relationships for the distributions of line sections in

. n / , B = I , R w , w = w.

These show how Z- and W-random line sections are preserved if the plane section is taken with the appropriate randomness. This result holds for compact domains, which do not have to be spherical, and also in higher dimensions. Davy and Miles (1977) prove a theorem of which the above is a special case for a UR g-flat across the section made by a UR r-flat through an n-dimensional compact domain.

There are many relationships given by mixtures of distributions. Some are obvious, such as

ns,w = n p , p = P(S) + f (W),

and the way that the probability density functions are presented in Table 7 goes some way in demonstrating them. For example, within the class of distributions Pk having probability density function 2(k + l ) zZk+' ( k = 0, 1,2) we have

zy n,,J = aw,, = Po,

Ri,rim PI, Po = &(a) + 4P*, 0 - N T , / + f P , ,

I - &,I. + kP2, I - 3ani,nT + 4P2,

p = -

p = 5

p = z

4T + iP, = $Po + iP*, 5n,,/ + %PI = $Po + BP2,

An,,, + &P, 3 $Po + 3P*.


TABLE 7: Distributions of the distance Z from the centre of a sphere of unit radius to the midpoint of a chord under the uniform randomness f l ~ ~ , ~ , ,

Z

U -

8 z n2 u

E - -

(t) + a (3zu)

[75 (:) + 35(3zu) + 21(5zu’) + 25(7zu5)

1 [ 11025 (:) + 2100(3zu) + 1134(5zu3) + 900(7zu5) 16384 I + 1225(9zu7)

3 z 3 3 z 1 2 u - - = 5 (--) - 5 (320)

2 2 _ _ n u

4 z2 - _ n u

- ; (; _ - Z ) + a (;zur)

1 5

32 3P2

9 8

75 64

1225 1024

-

-

-

-

19845 16384 -

8 3 n flw.ns -

3 - n

16 5 a

24 7 n

-

-

a2 8 - I ZK I

B 22

3 W, P TZE

5 - z{2( I + u2)E - u’K) pa 8

l “p - z{(8 + 7u2 + 8u4)E - 4(1 + u2)u2K} 48

4 3

97r2 64

- 1

-

75n2 512 -

1225~’ 8192 -

3 1 9 8 4 5 ~ ~ - 128 131072 - z{8(1 + u2)(6 - U’ + 6u4)E - (24 + 23d + 24u4)o2K}

I 3z(K - E) = ~ ( z K ) - 2 31r2 32 -


TABLE 7: (continued)

R X RY PZ(Z) EL R Z

3 8 -a n, 7 flI. W rIw 3z’

2 1 - (3z2) + - (6zu’r) na 3 3

5 9 n, 4 (32’) - 20 (5z4) +

2 5 - n

3 7 -71

16 - ZUK m2

128 9a2 -

3 - w, p3zu 7 W

3 8 8 5 (3zu) + - (5zu3)

1 - { 3 5 ( 3 ~ ~ ) + 14(5zu3) + 15(7zd)} 64

1 - {525(3zU) + l89(5zu3) + 135(7&) + 175(9zu7)} 1024

6z(1 - U) = 3(2z) - 2 ( 3 z ~ )

4 z2 - - 7 1 u

8 - z sin-’ z 71

12 - zur 71

16 - z2u 71

3 2 7 1 (‘6 zzu) - ; (: z4u)

L

25 16 -

1225 768 -

6615 4096

8 371 ow, -

32 971 f l L B -

4 1 4 z(2 - ZZ) = - (22) - - (4z’) T, Tu I 3 3 3

64 371’ - ZUE

W , P 4zu2 = 4z(1 - z2) = 2(2z) - ( 4 ~ ~ )

64 45 -

2048 13.571’

8 5

~

- a


TABLE 7: (continued)

Rx RY PZ(Z) EL RZ

5zu3

1 2 ( 1 + u2)u3z = - {7(5zu3) i- 5(7zuS)} 12 12

1 - {63(5zu3) + 30(7zd) + 35(9zu7)} 128

4z3

5 3

245 I44

-

-

44 1

256 -

128 45P

512 1 3 5 ~

64 157r

-

__

-

1024 225a

512 1057

__

-

22 1 - (4 - 42’ + 3z4) = - {8(2~) - 4(4z3) + (6~’)) 5 5

128 15n2

1 5 {12(2~) - 9(4Z3) + 2(6z5)}

T i I

- zu{2( 1 + u2)E - u2KJ

W, p

Pa 6zu4

a@ 7zu5

63 1 16 16

6 6 1 - z3(4 - zz) = - (427 - - (6~’) 5 5 5

- (1 + UZ)U5Z = - {9(7zu’) + 7(9zu7))

I

32 zz -- (15 - l0z2 + 3z4) 75n u n,

n, [5 (: uz2) - (; uz4) + 6 (: z sin-’ z)}

32 57r

64 - (u3z2 + u2z sin-’ z)

n, - uzZ(3 - z2)

na 5%

256 175

8192 5 2 5 ~ ’

288 175

-

__

-

12 7

T -

I - 4

567 320

192 175

1536 5 2 5 ~

2048 5257r

768 1757

4096 875%

-

-

-

-

-

~


TABLE 7: (concluded)

2 zr

8 - z sin-' z 71

3z2

5 1 - z2(3 - 2') = - {5(3z2) - ( 5 ~ ~ ) ) 4 4

- {35(3z2) - 14(5z4) + 3(7z6)) 1

24

1 - { 105(3z2) - 63(5z4) + 27(7z6) - 5(9zS)} 64

6z(r - z) = 3(2zr) - 2(3z2)

2 - - (2z) = 2z(u-- ' - 1) (3 -42 In v

6z(1 - U) = 3(2z) - 2(3zu)

42'

3 1 2 2 3z3(2 - z2) = - (4Z3) - - (6~ ' )

6144 1225n -

n %n, -

3

32 9 n n W, n, -

25 64 - n

1225 3072 " 6615

-

16384" " - 4

16 15 a , I -

8 7 -

APPENDIX. THE CHORD THROUGH TWO POINTS INDEPENDENTLY UNIFORMLY AT RANDOM IN A UNIT DISK

Let the two points Ql and Q2 have polar coordinates (TI, 41) and (r2, +2) with respect to an origin at the centre of a unit disk. Let and 42 be UR on [0,27r), and rl and r2 have probability density functionspl(rl) andp2(r2), with rl, r2, and 42 mutually independent.

Let the midpoint M of the chord through Ql and Q2 have polar coordinates ( x , e), and the signed distances of Ql and Q2 from M be yl and y 2 . Then when we transform from coordinates (rlr 41, r2, 42) to (x,y1,y2, O ) , their joint probability density function is

where 1 y 2 - yl I/rl r2 is the Jacobian and

r: = x2 + yf , r j = x2 + y;.

COLEMAN Vol. 17, No. 1

TABLE 8: Uniform randomness for a point in a unit disk and the probability density function p( r ) for the distance of the point from the centre of the disk

UR Description p ( r ) ( O s r s l )

I

C

A1

The point is UR in the interior of the disk. 2r

6 The point is UR on the circumference of the disk.

The point is the orthogonal projection onto a fixed diametral disk of a point which is UR in the interior of a unit sphere.

The point is the orthogonal projection onto a fixed diametral disk ofa point which is UR on the surface of a unit sphere.

3r(I - r2)b

Ac r( l - r 2 ) - *

TABLE 9: The distribution of the distance X from the centre of a unit disk to the midpoint of a uniformly random chord

through two independent points Q, and Q2

I I 16 - v3 T 371

I

4 - u TP 71

16 371 - u3 T,

3 - uz 2

- "4

8

4 - v a

1 ( = I )

71

C C P

We integrate out from (1 1) in succession with respect to 0, y2, and yl, to obtain the marginal probability density function for X, the distance from the centre of the disk to the chord, given in Table 9. The distributions used for p i ( r ) are given in Table 8.

The same calculations allow us to take the points Ql or Q2 to be on the circumference of the disk by taking p i ( r ) to be a probability density function having unit mass at r = 1 [a delta function, denoted 6 in Table 8, which can be thought of as the limit as u2 + 0 of the N(1, u2) probability density function].

In Table 9, u = m. The results are those of Table 2 , except for $u2 and 9 u 4 , which were not encountered there.


ACKNOWLEDGEMENT I am pleased to acknowledge an enjoyable visit to the University of Calgary in March and April

1987 as a Visiting Scholar, with field trips to Simon Fraser University and the Universities of British Columbia and Victoria, and Okanagan College. In particular I thank Dr. Enns and Dr. P.F. Ehlers for pleasurable collaboration and hospitality.

REFERENCES Bertrand, J. (1888). Calcul des probabilite's. Gauthier-Villars. Paris. Coleman, R. (1979). An introduction to mathematical stereology. Memoirs no. 3. Dept. of Theoretical Statistics,

Coleman, R. (1981). The construction of invariant random paths through three-dimensional specimens. J.

Coleman, R. (1987a). Line section sampling of Wicksell's corpuscles. Acta Stereol., 6(1), 33-36. Coleman, R . (1987b). Stereological estimation of mineral liberation. I. Concentric sphere models. Actu Stereol.,

Davy, P., and Miles, R.E. (1977). Sampling theory for opaque spatial specimens. J. Roy. Sfatisf. SOC. Ser. B ,

Ehlers, P.F., and Enns, E.G. (1981). Random secants of a convex body generated by surface randomness. J.

Enns, E.G., and Ehlers, P.F. (1978). Random paths through aconvex region. J. Appl. Probab., 15, 144-152. Ganvood, F., and Holroyd, E.M. (1966). The distance of a "random chord" of a circle from the centre. Mafh.

Jaynes, E.T. (1973). The well-posed problem. Found. Phys., 3,477-492. Kendall, M.G., and Moran, P.A.P. (1963). Geomefrical Probability. Griffin, London. Kingman, J.F.C. (1969). Random secants of a convex body. J. Appl. Probab., 6,660-672. Lehmann, E.L. (1985). The Neyman-Pearson lemma. Encyclopedia of Statistical Sciences. Volume 6 ( S . Kotz

and N.L. Johnson, Eds.) , Wiley, New York, 224-230. Miles, R.E. (1971). Isotropic random simplices. Adv. Appl. Probab., 3, 353-382. Solomon, H. (1978). Geometric Statistics. SIAM, Philadelphia. Streit, F. (1978). On a statistical approach to Bertrand's problem. Elem. Math., 33(6), 134-138. Wicksell, S.D. (1925). The corpuscle problem. A mathematical study of a biometric problem. Biomefrika, 17,

Institute of Mathematics, University of Aarhus, DK-8000 Aarhus C, Denmark.

Microsc., 122, 105-106.

6(2), 185-192.

39, 56-65.

Appl. Probab., 18,157-166.

Guz., 50,283-286.

84-99.

Received 4 August 1987 Revised 24 November 1988 Accepted 28 November 1988

Department of Mathematics Imperial College

London SW7 282 England.

random sections of a sphere

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