Surface Area and Capacity of Ellipsoids in N Dimensions

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NEW ZEALAND JOURNAL OF MATHEMATICS Volume 34 (2005), 165?198

SURFACE AREA AND CAPACITY OF ELLIPSOIDS IN n DIMENSIONS

Garry J. Tee

(Received March 2004)

Abstract. The surface area of a general n-dimensional ellipsoid is represented as an Abelian integral, which can readily be evaluated numerically. If there are only 2 values for the semi-axes then the area is expressed as an elliptic integral, which reduces in most cases to elementary functions. The capacity of a general n-dimensional ellipsoid is represented as a hyperelliptic integral, which can readily be evaluated numerically. If no more than 2 lengths of semi-axes occur with odd multiplicity, then the capacity is expressed in terms of elementary functions. If only 3 or 4 lengths of semi-axes occur with odd multiplicity, then the capacity is expressed as an elliptic integral.

1. Introduction

Adrien-Marie Legendre published in 1788 a convergent series for the surface area of a general ellipsoid [Legendre 1788], and in 1825 he published an explicit expression for that area in terms of his standard Incomplete Elliptic Integrals [Legendre 1825]. But Legendre's results remained very little-known, and several authors (e.g. [Keller]) have published assertions that there is no known formula for the surface area of a general ellipsoid. Derrick Lehmer constructed [Lehmer] a series expansion for the surface erea of an n-dimensional ellipsoid, which differs from Legendre's series when n = 3.

Philip Kuchel and Brian Bulliman studied surface area of red bloodcells, which they modelled by ellipsoids, and they constructed a series expansion (different from Legendre's) for the surface area [Kuchel & Bulliman]. Leo Maas studied locomotion of unicellular marine organisms, which he modelled by ellipsoids, and he used Legendre's expression for the area [Maas]. Igathinathane and Chattopadhyay studied the skin of rice grains, which they modelled by ellipsoids, and they constructed tables for the surface area [Igathinathane & Chattopadhyay]. Reinhard Klette and Azriel Rosenfeld developed algorithms for computing surface area of bodies from discrete digitizations of those bodies, and they tested their software on digitized ellipsoids, comparing the result of their algorithm with the surface area evaluated by numerical integration [Klette & Rosenfeld].

The electrostatic capacity of an ellipsoid has been known since the 19th century [P?olya & Szego].

1991 Mathematics Subject Classification primary 41A63 41A55, secondary 41-03 01A50 01A55. Key words and phrases: ellipsoid, n dimensions, surface area, capacity, Legendre, elliptic integral, hyperelliptic integral, Abelian integral.

166

GARRY J. TEE

For n-dimensional ellipsoids, Bille Carlson constructed upper and lower bounds for the surface area and for the electrostatic capacity [Carlson 1966]. But no numerical values for either surface area or for capacity appear to have been published, for any ellipsoid in more than 3 dimensions.

This paper contructs definite integrals for surface area and for capacity of ndimensional ellipsoids, and several numerical examples are computed in up to 256 dimensions.

2. Spheroids and Ellipses

Consider an ellipsoid centred at the coordinate origin, with rectangular Cartesian coordinate axes along the semi-axes a, b, c,

x2 y2 z2

a2 + b2 + c2 = 1.

(1)

2.1. Surface area of spheroid.

In 1714, Roger Cotes found the surface area for ellipsoids of revolution [Cotes],

called spheroids.

For the case in which two axes are equal b = c, the surface is generated by

rotation

around

the

x?axis

of

the

half?ellipse

x2 a2

+

y2 b2

= 1 with y 0. On that

half-ellipse, dy/dx = -b2x/(a2y), and hence the surface area of the spheroid is

a

A = 2 2y

0

b4x2

a

1 + a4y2 dx = 4 0

y2

+

b4 a4

x2

dx

a

x2 b2 x2

= 4b

0

1 - a2 + a2 a2 dx

1

= 4ab

0

b2 1 - 1 - a2

1

u2 du = 4ab

0

1 - u2 du,

(2)

where u = x/a and = 1 - b2/a2. Therefore, the surface areas for prolate spheroids

(a > b), spheres (a = b) and oblate spheroids (a < b) are:

arcsin

2b

a?

+b

(prolate),

A = 2b(a + b) = 4a2

(sphere),

(3)

2b

arcsinh a?

- +b

(oblate) .

-

Neither the hyperbolic functions nor their inverses hadthen been invented, and

Cotes gave a logarithmic formula for the oblate spheroid [Cotes, pp. 169-171]. In

modern notation [Cotes, p.50],

A = 2a2 + b2 1 log 1 + - .

(4)

- 1 - -

For || 1, either use the power series for (arcsin x)/x to get

A

=

2b

a

1

+

1 6

+

3 40

2

+

5 112

3

+

???

+b

,

(5)

ELLIPSOIDS IN n DIMENSIONS

167

or else expand the integrand in (2) as a power series in u2 and integrate that term by term:

1

A = 4ab

1 - u2 1/2du

0

1

1 -1

1 -1 -3

= 4ab

0

1

-

1 2

u2

+

22

2!

2u4 -

2

22

3!

3u6

1 -1 -3 -5

1 -1 -3 -5 -7

+ 2 2 2 2 4u8 - 2 2 2 2 2 5u10 + ? ? ? du

4!

5!

1 1 2 1 3 5 4 7 5

= 4ab 1 - - -

-

-

-??? .

(6)

2 3 8 5 16 7 128 9 256 11

2.2. Circumference of ellipse.

In 1742, Colin MacLaurin constructed a definite integral for the circumference of

an ellipse [MacLaurin]. Consider an ellipse with semi-axes a and b, with Cartesian

coordinates along the axes:

x2 y2

a2 + b2 = 1 ,

(7)

On that ellipse, 2x dx/a2 + 2y d y/b2 = 0, and hence dy/dx = -b2x/(a2y), and the

circumference is 4 times the ellipse quadrant with x 0 and y 0. That quadrant

has arclength

a

b4x2

a

b2(x/a)2

I=

0

1 + a4y2 dx = 0

1 + a2(y/b)2 dx

a

(b/a)2(x/a)2

=

0

1 + 1 - (x/a)2 dx .

(8)

Substitute z = x/a, and the circumference becomes

1

(b/a)2z2

1 1 - mz2

4I = 4a

0

1 + 1 - z2

dz = 4a

0

1 - z2 dz ,

(9)

where

b2

m = 1 - a2 .

(10)

With a b this gives 0 m < 1.

That integral could not be expressed finitely in terms of standard functions.

Many approximations for the circumference L(a, b) of an ellipse have been pub-

lished, and some of those give very close upper or lower bounds for L(a, b) [Barnard,

Pearce & Schovanec]. A close approximation was given by Thomas Muir in 1883:

L(a, b) M (a, b)

d=ef

2

a3/2 + b3/2

2/3

.

(11)

2

That is a very close lower bound for all values of m (0, 1). Indeed, [Barnard, Pearce & Schovanec, (2)]:

0.00006m4 < L(a, b) - M (a, b) < 0.00666m4 .

(12)

a

168

GARRY J. TEE

2.3. Legendre on elliptic integrals.

Adrien-Marie Legendre (1752-1833) worked on elliptic integrals for over 40 years, and summarized his work in [Legendre 1825]. He investigated systematically the integrals of the form R(t, y) dt, where R is a general rational function and y2 = P (t), where P is a general polynomial of degree 3 or 4. Legendre called them "fonctions ?elliptique", because the formula (9) is of that form -- now they are called elliptic integrals. He shewed how to express any such integral in terms of elementary functions, supplemented by 3 standard types of elliptic integral.

Each of Legendre's standard integrals has 2 (or 3) parameters, including x = sin . Notation for those integrals varies considerably between various authors. Milne?Thomson's notation for Legendre's elliptic integrals [Milne?Thomson, ?17.2] uses the parameter m, where Legendre (and many other authors) had used k2.

Each of the three kinds is given as two integrals. In each case, the second form is obtained from the first by the substitutions t = sin and x = sin .

The Incomplete Elliptic Integral of the First Kind is:

F (|m) d=ef

d

x

=

dt

. (13)

0 1 - m sin2

0 (1 - t2)(1 - mt2)

The Incomplete Elliptic Integral of the Second Kind is:

E(|m) d=ef

0

1 - m sin2 d =

x 0

1 - mt2 1 - t2 dt .

(14)

That can be rewritten as

x

1 - mt2

dt ,

(15)

0 (1 - t2)(1 - mt2)

which is of the form R(t, y) dt, where y2 = (1 - t2)(1 - mt2). The Incomplete Elliptic Integral of the Third Kind is:

(n; |m) d=ef

d

0 (1 - n sin2 ) 1 - m sin2

x

dt

=

.

(16)

0 (1 - nt2) (1 - t2)(1 - mt2)

The

special

cases

for

which

=

1 2

(and

x

=

1)

are

found

to

be

particularly

important, and they are called the Complete Elliptic Integrals [Milne?Thomson,

?17.3].

The Complete Elliptic Integral of the First Kind is:

K (m)

d=ef F

1 2

|m

d=ef

/2

d

0

1 - m sin2

1

dt

=

.

(17)

0 (1 - t2)(1 - mt2)

The Complete Elliptic Integral of the Second Kind is:

E(m)

d=ef

E

1 2

|m

=

/2 0

1 - m sin2 d =

1 0

1 - mt2 1 - t2 dt .

(18)

ELLIPSOIDS IN n DIMENSIONS

169

The complete elliptic integrals K(m) and E(m) can efficiently be computed to high precision, by constructing arithmetic-geometric means [Milne?Thomson. ?17.6.3 & 17.6.4].

3. Surface Area of 3-Dimensional Ellipsoid

For a surface defined by z = z(x, y) in rectangular Cartesian coordinates xyz, the standard formula for surface area is:

z 2 z 2

Area =

1+

+

dx dy.

(19)

x

y

On the ellipsoid (1),

z -c2x

z -c2y

= x

a2z ,

= y

b2z .

(20)

Consider the octant for which x, y, z are all non?negative. Then the surface

area for that octant is

a b 1-x2/a2

S=

c4x2 c4y2 1 + a4z2 + b4z2 dy dx

0

0

=

a

b 1-x2/a2

z2

sqrt c2

+

c2 x2 a2 a2 +

z2/c2

c2 b2

y2 b2

dy dx

0

0

a b 1-x2/a2

=

0

0

x2 y2 c2 x2 c2 y2

1 - a2 - b2 + a2 a2 + b2 b2 x2 y2

dy dx

1 - a2 - b2

a b 1-x2/a2

=

0

0

c2 x2

c2

1 - 1 - a2 a2 - 1 - b2

x2 y2 1 - a2 - b2

y2 b2

dy dx .

(21)

Hence, if two semi-axes (a and b) are fixed and the other semi-axis c increases,

then the surface area increases.

Denote

c2

c2

= 1 - a2 ,

= 1 - b2 ,

(22)

and then (21) becomes

S=

a b 1-x2/a2

0

0

x2 y2

1 - a2 - b2 x2 y2

dy dx .

1 - a2 - b2

(23)

For a general ellipsoid, the coordinate axes can be named so that a b c > 0, and then 1 > 0.

170

GARRY J. TEE

3.1. Legendre's series expansion for ellipsoid area.

In 1788, Legendre converted this double integral to a convergent series [Legendre

1788] [Legendre 1825, pp. 350?351].

Replace the variables of integration (x, y) by (, ), where cos = z/c, so that

x2 a2

+

y2 b2

= sin2 ,

or

x2

y2

(a sin )2 + (b sin )2 = 1 .

(24)

Then let cos = y/(b sin ) so that sin = x/(a sin ), or

x = a sin sin , y = b sin cos .

(25)

Differentiating x in (25) with respect to (with constant ), we get that dx =

a

cos

sin

d;

and

differentiating

the

equation

x2 a2

+

y2 b2

=

sin2

with

respect

to

(with constant x), we get that 2y dy = 2b2 sin cos d. Thus the element of area

in (23) becomes

dx dy = ab sin cos d d,

(26)

and the area S of the ellipsoid octant becomes [Legendre 1825, p.350]

/2 /2 =0 =0

1 - sin2

sin2 -

sin2

-

x2 a2

1 - sin2

ab sin cos d d

/2 /2

= ab

sin 1 - ( sin2 + cos2 ) sin2 d d . (27)

=0 =0

Thus,

/2 /2

S = ab

sin

=0 =0

1 - p sin2 d d ,

(28)

where p is a function of :

p = sin2 + cos2 = + ( - ) sin2 .

(29)

Hence,

as

increases

from

0

to

1 2

,

p

increases

from

0

to

< 1.

Define

/2

I(m) d=ef

sin 1 - m sin2 d .

(30)

0

Clearly, I(m) is a decreasing function of m (for m 1). That integral can be expressed explicitly. For m (0, 1),

1 1-m 1+ m

I(m) = + log

.

(31)

2 4m

1- m

ELLIPSOIDS IN n DIMENSIONS

171

Expand the integrand in (30) as a power series in p and integrate for from 0

to

1 2

,

to

get

a

series

expansion

for

I (p):

/2

I(p) =

sin (1 - p sin2 )1/2 d

=0

/2

1 -1

1 -1 -3

=

0

sin

1

-

1 2

p

sin2

+

22

2!

p2 sin4 -

2

22

3!

p3 sin6 + ? ? ?

/2

/2

/2

=

sin d

-

1 2

p

sin3 d

-

1?1 2?4

p2

sin5 d

0

0

0

/2

/2

-

1?1?3 2?4?6

p3

sin7 d

-

1?1?3?5 2?4?6?8

p4

sin9 d - ? ? ? .

0

0

d (32)

Define

? 1 ? 3 ? 5 . . . (k - 1)

sk d=ef

/2

sink d

=

2?2?4?6...k

, (even k 2) , (33)

0

2 ? 4 ? 6 . . . (k - 1) ,

(odd k 3) ,

3?5?7...k

with

s0

=

1 2

and

s1

=

1.

For

all

k

>

-1

[Dwight,

?854.1],

sk

=

2

1 2

(1

+

k)

1

+

1 2

k

.

(34)

In particular,

s1 = 1, s3

=

2 3

,

s5

=

2?4 3?5

,

s5

=

2?4?6 3?5?7

,...

,

(35)

and hence

I (p)

=

1

-

1 1?3

p

-

1 3?5

p2

-

1 5?7

p3

-

1 7?9

p4

-

???

.

(36)

Therefore, the surface area of the ellipsoid is

/2

A = 8ab

I(p) d

0

/2

= 8ab

0

1

-

1 1?3

p

-

1 3?5

p2

-

1 5?7

p3

-

1 7?9

p4

-

???

d

= 4ab

1-

1 1?3

P1

-

1 3?5

P2

-

1 5?7

P3

-

1 7?9

P4

-

?

?

?

d ,

where

P1

=

2

/2

( sin2 + cos2 ) d =

0

1 2

+

1 2

,

P2

=

2

/2

( sin2 + cos2 )2 d =

0

1?3 2?4

2

+

1?1 2?2

+

1?3 2?4

2,

P3

=

2

/2

( sin2 + cos2 )3 d

0

=

1?3?5 2?4?6

3

+

1?3?1 2?4?2

2

+

1?1?3 2?2?4

2

+

1?3?5 2?4?6

3,

et cetera.

(37) (38)

172

GARRY J. TEE

Legendre gave [Legendre 1825, p.51] a generating function for the Pk:

1

= (1 - z)-1/2(1 - z)-1/2

(1 - z)(1 - z)

=

1

+

1 2

z

+

1?3 2?4

2z2

+

1?3?5 2?4?6

3

z3

+

???

1

+

1 2

z

+

1?3 2?4

2

z2

+

???

= 1 + P1z + P2z2 + P3z3 + P4z4 + ? ? ? .

(39)

Infinite series had been used by mathematicians since the 13th century in India and later in Europe, but very little attention had been given to convergence. Consequently much nonsense had been published, resulting from the use of infinite series which did not converge. From 1820 onwards, Cauchy developed the theory of infinite series, and he stressed the importance of convergence [Grabiner, Chapter 4]. In 1825, Legendre carefully explained that his series (37) for the area does converge [Legendre 1825,p.351].

All terms after the first in Legendre's series (37) are negative, and hence the partial sums of that series decrease monotonically towards the surface area.

I have searched many books on elliptic integrals and elliptic functions, and I have not found any later reference to Legendre's series (37) for the surface area of a general ellipsoid.

Derrick H. Lehmer stated (in 1950) a different infinite series for the surface area, in terms of the eccentricities

b2

c2

= 1 - a2 . = 1 - a2 .

(40)

The surface area is

S(a, b, c)

=

4ab

1 1-

2 + 2

1

34 + 222 + 34 - ? ? ?

6

120

()

2 + 2

= 4ab 1 - 42 P 2

,

(41)

=0

where P(x) is the Legendre polynomial of degree [Lehmer, (6)]. Philip Kuchel and Brian Bulliman constructed (in 1988) a more complicated

series expansion for the area [Kuchel & Bulliman].

3.2. Bounds for ellipsoid area.

As

increases

from

0

to

1 2

,

then

sin2 +

cos2

=

( - ) sin2 +

increases

from to . Hence, for all values of , the integrand in (27) lies between the upper

and lower bounds

sin 1 - sin2

sin 1 - ( sin2 + cos2 ) sin2 sin 1 - sin2 . (42)

Accordingly, for all values of , the integral over in (27) lies between the upper and lower bounds

/2

I()

sin 1 - ( sin2 + cos2 ) sin2 d I() . (43)

=0

................
................

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