Since there does not appear to be a complete derivation of the surface area of an ellipsoid readily available on the web, consider the manual evaluation in all its lovely detail. The concept of volume element indicates that surface area can be calculated much like volume, as an integral over the square root of the determinant of the metric, and that is how the evaluation will proceed.

A typical parametrization of an ellipsoid is

r=[ asinθcosφ, bsinθsinφ, ccosθ]

Forming the space-like interval on this surface, the corresponding metric tensor is

[ (a2 cos2φ +b2 sin2φ) cos2θ +c2 sin2θ (b2 -a2) sinθcosθ sinφcosφ (b2 -a2) sinθcosθ sinφcosφ (a2 sin2φ +b2 cos2φ) sin2θ ]

which looks rather complicated, but the determinant simplifies significantly:

detg =a2b2 sin2θ cos2θ +c2 (a2 sin2φ +b2 cos2φ) sin4θ

Evaluating an integral of the square root of this expression can be done with expansions of binomials. The overall factor of a sine in the square root of this determinant makes the choice x=cosθ natural for reducing factors in the integrand, and the other angular variable can be left in place. With the abbreviations

β=1 -a2 b2 γ=1 -a2 c2

and taking advantage of symmetries in the integrand, the surface area integral is

S=0π dθ 02π dφ detg =8bc 01 dx 1-γx2 0π/2 dφ 1-β 1-x2 1-γx2 sin2φ

Given the generalized binomial expansion

(1-x )r =1-rx +r (r-1) 2! x2 -r(r-1) (r-2) 3! x3 + (1-x )r =k=0 Γ(r +k) Γ(r) xkk!

as well as a trigonometric integral, the integration over φ is

0 π/2 dφ 1-β 1-x2 1-γx2 sin2φ =0 π/2 dφ k=0 Γ(12 +k) Γ(12 ) βk k! (1-x2 1-γx2 )k sin2kφ =Γ( 12)2 k=0 Γ(12 +k) Γ(12 ) Γ(k +12) Γ(k+1) βk k! (1-x2 1-γx2 )k =π2 k=0 Γ(12 +k) Γ(12) Γ(12 +k) Γ(12 ) βk (k!)2 (1-x2 1-γx2 )k

As an aside, this is the expansion of a complete elliptic integral of the second kind. With one more application of the binomial expansion, the surface area becomes

S=4πbc 01 dx 1-γx2 k=0 Γ(12 +k) Γ(12 ) Γ(12 +k) Γ(12 ) βk (k!)2 (1-x2 1-γx2 )k S =4πbc 01 dx k,l=0 Γ(12 +k) Γ(12 ) Γ(12 +k) Γ(12 ) Γ(k- 12+l) Γ(k- 12) βk γl (k!)2 l! x2l (1-x2 )k S=4πbc k,l=0 Γ(12 +k) Γ(12 ) Γ(k- 12+l) Γ(12 ) βk γl (k!)2 l! 01 dx x2l (1-x2 )k

The last binomial has not been expanded because now one can take advantange of a binomial integral,

01 dx x2l (1-x2 )k =12l+1 F12 (k, l+12; l+32; 1) =12l+1 Γ(l +32) Γ(k+1) Γ(k+l +32) Γ(1) =Γ(l +12) Γ(12) Γ(32) Γ(k+l +32) k!

where the last line uses the Gauss summation formula. Putting this into the previous sum and reordering arguments slightly gives

S=4πbc k,l=0 Γ(12 +k+l) Γ(12 ) Γ(12 +k) Γ(12) Γ(12 +l) Γ(12) Γ(32) Γ(32 +k+l) βkk! γll! S =4πbc F1( 12; 12, 12; 32; β,γ) S=4πbc F1( 12; 12, 12; 32; 1-a2 b2, 1-a2 c2)

which is a single Appell double hypergeometric function. This single Appell function form for the surface of an ellipsoid does not seem to be widely known. While a should be less than both b and c in accordance with the intermediate elliptic integral expansion, this form does not impose an ordering on b and c due to symmetry in the second and third parameters.

For a symmetric ellipsoid with a=b the surface area reduces to a Gauss hypergeometric function that can be related to circular functions,

Sa=b =4πac F12 (12, 12; 32; γ) Sa=b =4πac[ 12 F12 (12, 12; 32; γ) +12 F12 (12, 12; 12; γ)] Sa=b =2πac[ sin1 γγ +1-γ]

where the second step applies a contiguous relation and the third identifies two special cases. The two options for parameters are

Sprolate =2πa2 +2πac 1-a2 c2 sin1 1-a2 c2 , a<c Soblate =2πa2 +2πac a2 c2 -1 sinh1 a2 c2 -1 , a>c

using the fact that the inverse sine or inverse hyperbolic sine of an imaginary argument is the other function multiplied by the imaginary unit. For ca both expressions simplify to

Ssphere =4πa2

since both the inverse sine and and inverse hyperbolic sine have a leading linear term in their expansions.

To relate the single Appell function form to Legendre elliptic integrals, recall the expansions near the beginning of this presentation:

F(φ|m) =sinφ F1( 12; 12, 12; 32; sin2φ, msin2φ) E(φ|m) =sinφ F1( 12; 12, 12; 32; sin2φ, msin2φ)

The result for the surface area can be expressed as a linear superposition of the Appell functions appearing here, along with a third function to be determined:

F1( 12; 12, 12; 32; β,γ) =AF1( 12; 12, 12; 32; β,γ) +BF1( 12; 12, 12; 32; β,γ) +f(β, γ)

Since the only one of these three Appell functions not symmetric in the second and third parameters is the third, its minus sign will be associated with the argument β for final consistency. The simplest way to establish the linear superposition appears to via an integral representation of the Appell function:

F1( a; b1, b2; c; z1, z2) =Γ(c) Γ(a) Γ(c-a) 01 dx xa-1 (1-x) c-a-1 (1-z1x )b1 (1-z2x )b2

While this representation does not converge at the lower limit for the cases of parameters considered here, one can assume the infinities cancel from both sides of the equation and proceed anyway. With identifications

Γ(32) =12 Γ(12) =14 Γ(12 )

the linear superposition takes the form

14 01dx (1-x) x3/2 1-βx 1-γx =A2 01dx 1x 1-βx 1-γx +B2 01dx 1-βx x 1-γx +f(β, γ)

Moving integrals to the left-hand side of the equation, the combined integrand is

12x 1-βx 1-γx [1-x 2x +A +B(1 -βx)]

Since derivatives of linear square roots have one less power algebraically, a combination to consider is

ddx 1-βx 1-γx x =12x 1-βx 1-γx (1x -βγx)

For the term inverse in x here to match that in the previous bracket, an overall factor of one half is needed and the other constants follow immediately:

12 (1x -βγx) =1-x 2x +A +B(1 -βx) A=1-γ 2 , B=γ2

The remaining function, ignoring the singular lower limit, is now simply

f(β,γ) =12 01dx ddx 1-βx 1-γx x =12 1-β 1-γ

and the linear relation among the three Appell functions is

F1( 12; 12, 12; 32; β,γ) =1-γ2 F1( 12; 12, 12; 32; β,γ) +γ2 F1( 12; 12, 12; 32; β,γ) +12 1-β 1-γ

With the identifications

β=msin2φ γ=sin2φ 1-γ=cos2φ

the surface area of the three-dimensional ellipsoid can be written

S=2πa2 +2πbc sinφ [cos2φ F(φ|m) +sin2φ E(φ|m)]

which is the expected result with a and c interchanged. This form reduces to the result for the prolate ellipsoid for a=b , which for the traditional Legendre form means m=0 .

The single Appell function result above is patently simpler than the traditional form.

Using a relation of the Carlson completely symmetric integral second kind of the to Legendre elliptic integrals

RG (cos2φ, 1-msin2φ, 1) =cos2φ 2sinφ F(φ|m) +sinφ2 E(φ|m) +cosφ2 1-msin2φ

which for the given parameter choices becomes

RG (1-γ, 1-β, 1) =S -2πa2 4πbc +12 1-γ 1-β =S 4πbc

the surface area of the ellipsoid can be written

S=4πbc RG( a2 b2, a2 c2, 1)

using the symmetry of the arguments of the Carlson elliptic integral. This expression has the same overall factors as the single Appell function form, so that this particular Carlson function is identical to the single Appell function.

Uploaded 2020.06.21 — Updated 2020.07.13