Surface Area and Capacity of Ellipsoids in N Dimensions
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 ndimensional ellipsoid is represented as an Abelian integral, which can readily be evaluated numerically. If there are only 2 values for the semiaxes then the area is expressed as an elliptic integral, which reduces in most cases to elementary functions. The capacity of a general ndimensional ellipsoid is represented as a hyperelliptic integral, which can readily be evaluated numerically. If no more than 2 lengths of semiaxes occur with odd multiplicity, then the capacity is expressed in terms of elementary functions. If only 3 or 4 lengths of semiaxes occur with odd multiplicity, then the capacity is expressed as an elliptic integral.
1. Introduction
AdrienMarie 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 littleknown, 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 ndimensional 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 4103 01A50 01A55. Key words and phrases: ellipsoid, n dimensions, surface area, capacity, Legendre, elliptic integral, hyperelliptic integral, Abelian integral.
166
GARRY J. TEE
For ndimensional 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 semiaxes 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
halfellipse, 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. 169171]. 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 semiaxes 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.
AdrienMarie Legendre (17521833) 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 arithmeticgeometric means [Milne?Thomson. ?17.6.3 & 17.6.4].
3. Surface Area of 3Dimensional 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 1x2/a2
S=
c4x2 c4y2 1 + a4z2 + b4z2 dy dx
0
0
=
a
b 1x2/a2
z2
sqrt c2
+
c2 x2 a2 a2 +
z2/c2
c2 b2
y2 b2
dy dx
0
0
a b 1x2/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 1x2/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 semiaxes (a and b) are fixed and the other semiaxis c increases,
then the surface area increases.
Denote
c2
c2
= 1  a2 ,
= 1  b2 ,
(22)
and then (21) becomes
S=
a b 1x2/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 1m 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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