Somebody must have thought of it before, but this formula seems unpublished...
If you know better, please
tell me
and I'll post updates here.
Meanwhile, I shall refer to the above approximation as
Thomsen's formula (2004-04-26).
For a very long prolate ellipsoid of revolution,
Thomsen's expression is
21+1/p / p
times the true surface area
(a relative error of about -1.41544 %).
This formula is also discussed in the next article.
On 2004-05-13, Knud Thomsen
(Denmark) wrote: [edited summary]
The expression
S » 4p
[ ( apbp +
apcp +
bpcp ) / 3 ] 1/p
approximates the true surface area of the ellipsoid with the least relative error
(± 1.061 % worst case)
when p » 1.6075 [...]
Best regards, Knud Thomsen
|
This second form of the formula is exact for a sphere regardless of the value of p,
whereas the one we first gave is always exact for a degenerate ellipsoid
(c = 0).
Both coincide for the value p = lg(3) = 1.5849625...
first proposed by Thomsen.
Knud Thomsen (2004-05-14) and David Cantrell
(2004-05-16) independently mentioned
the obvious generalization to n-dimensional ellipsoids
(with semiaxes a1, ... an )
which may be expressed in terms of the Hölder mean (H)
of the n products of n-1 semiaxes
( H = 0 when two or more semiaxes are zero):
H =
(a1a2 ... an )
[ (a1-p + a2-p + ... +
an-p ) / n ] 1/p
[nonzero semiaxes]
H =
(a1a2 ...
an-1 ) / n 1/p
[when one semiaxis, say an, is zero]
The hypervolume (V)
and hyperarea (S)
of an n-dimensional hypersphere are:
V = Rn
pn/2 / G(1+n/2)
[R is the hypersphere radius]
S = 2 Rn-1
pn/2 / G(n/2)
= 2 H
pn/2 / G(n/2)
Scaling considerations translate into the exact formula for the hypervolume (V)
of an ellipsoid (replace Rn by
a1a2...an ).
The n-1 dimensional case gives half the hyperarea of the [two sided]
degenerate n-dimensional ellipsoid, which is thus:
S' = 2 ( n1/p H )
p[n-1]/2 / G(1+[n-1]/2)
The expressions for S (hypersphere) and S' (degenerate hyperellipsoid)
are thus identical functions of H (generalizing the
YNOT formula for the ellipse) when:
n1/p =
Öp G(n/2+1/2) / G(n/2)
p =
|
Log (n) |
 |
| Log (A) |
The quantity A =
Öp G(n/2+1/2) / G(n/2)
has different elementary expressions in terms of
double-factorials depending on the parity of n:
- If n = 2k, then
A = p (2k-1)!! / 2k (k-1)!
= (p/2) (n-1)!! / (n-2)!!
- If n = 2k+1, then
A = 2k k! / (2k-1)!!
= (n-1)!! / (n-2)!!
The limit of this value of p is 2 for [very] large values of n...
Thomsen and Cantrell both expressed some doubts about the future popularity
of such approximate formulas for the hyperareas of hyper-ellipsoids...
On 2004-05-14, David W. Cantrell
wrote: [edited summary]
I just now noticed [the recent update(s) to this page].
Congratulations, Knud, on your approximation of 2004-04-26
and your more recent one with p = 1.6075...
I would prefer p = 8/5 which makes the latter algebraic and optimal
for nearly spherical ellipsoids. The relative error is still never worse than
-1.178 %
Best regards, David W. Cantrell
|
A fast feedback from David, who posted
other
approximations shortly thereafter.
An ellipsoid of semiaxes a, b and c is
called scalene if these three quantities are distinct, or rather [using the
inclusive viewpoint which is almost always
preferred in a mathematical context] when no two of them are known to coincide...
Let a be the largest semiaxis and c the smallest one.
Let e =
Ö(1-c2/a2)
The surface area (S) of the ellipsoid has a simple expression in 3 special cases:
for an oblate or prolate ellipsoid of revolution,
and for a degenerate ellipsoid (namely, a flat spheroid
whose surface consists of the two sides of
an ellipse) :
- If a = b, then
S = 2p [ a2 +
c2 atanh(e)/e ]
Oblate ellipsoid
(M&M's).
- If b = c, then
S = 2p [ c2 +
ac arcsin(e)/e ]
Prolate ellipsoid (cigar).
- If c = 0, then
S = 2p ab
The above symmetrical formula,
proposed in 2004 by the Danish geologist Knud Thomsen, is exact
for either a sphere ( a = b = c ) or a flat spheroid.
It's to the ellipsoid area what the
YNOT formula
is to the ellipse perimeter.
Dropping the simplicity and symmetry of Thomsen's formula, it's fairly easy
to devise expressions that are correct in all three of the above trivial cases...
For example, we may define atanh(x)/x and arcsin(x)/x by continuity when x is zero
(both quantities are thus equal to 1 when x is 0) and consider the expression:
S »
2p [ a (b-c) +
ac arcsin(u)/u + c2 atanh(v)/v ]
where
u = Ö(1-b2/a2)
and
v = Ö(1-c2/b2)
This turns out to be a poor kind of "improvement" over
Thomsen's formula except, of course,
when the ellipsoid is a solid of revolution, or nearly so.
The worst discrepancy between the two expressions
occurs when a is much larger than the other two semiaxes and c
is lb, with
l = (p/2-1)1/(p-1)
which makes Thomsen's expression
(1+lp )1-1/p
times the other (» 7.578 % larger).
As Thomsen's value is already a -1½ % underestimate,
the above is -9 % off...
This goes to show that there's a price to pay for breaking up a natural symmetry
in an approximation (as asymmetrical error terms don't automatically cancel out).
This remark also applies to another approximation (< 2.1 %) proposed in 2003
by Dr. Andreas Dieckmann (of Bonn University) in terms of
r = arcsin(e)/e :
where
