Extreme-perfect approx. of ellipse perimeter
David W. Cantrell
much like "very unique clones"; rather, I'm talking about
approximations of the perimeter of an ellipse which are exact at the
extremes of eccentricity, that is, at e=0 and at e=1. For an ellipse
with semiaxes a(major) and b(minor) [Hmm...The keys for "Music of the
Ellipsoids" perhaps?], these approximations give exactly 2*pi*a when
b=a (e=0) and 4*a when b=0 (e=1).
It should be noted that the great majority of well known approximations
of the perimeter are designed to be accurate especially for _small_
eccentricities. Not surprisingly then, many of these approximations are
not very accurate for large eccentricities.
I am interested in approximation formulae which satisfy three criteria:
(A) Extreme-perfection,
(B) Simplicity, a subjective criterion, and
(C) Good accuracy between the extremes of eccentricity.
[Perhaps these should be condensed to just two criteria:
(A) Extreme-perfection and
(B') A high ratio of overall accuracy to complexity.]
Why am I interested in extreme-perfect formulae? Perhaps it's primarily
a matter of aesthetics. Ramanujan also seems to have been interested in
them. Besides his two well known approximation formulae, which are not
extreme-perfect, he gave two formulae which are extreme-perfect.
Although their overall accuracies are impressive, they are not well
known, presumably because of their complexities.
Are there _simple_ extreme-perfect approximation formulae "in the
literature"? Yes. There are two of which I am aware.
(1) Perimeter is approx. 4a(1 + (pi/2-1)(b/a)^p) where the power p is
determined numerically to be approx. 1.45548 (in order to minimize the
extreme of |relative error|). [Unfortunately I do not know who first
suggested an approximation in this form. Does anyone have that
information?] The formula is clearly extreme-perfect and is quite
simple. Furthermore, for all eccentricities,
|relative error| < 0.004421.
(2) Perimeter is approx. 4(a^y + b^y)^(1/y) where y = ln(2)/ln(pi/2).
This formula was first proposed by Roger Maertens, and was published
last year in an article he co-authored with Ronald Rousseau. It's easy
to verify that this so-called YNOT formula is extreme-perfect, and the
formula is simple. For 0<e<1, it always gives an overestimate of the
perimeter. Since maximum relative error is approx. 0.00362, the YNOT
formula is more accurate than formula (1) above.
It's interesting to note that the YNOT approximation is similar to two
other well known approximations,
that of Euler (1773): 2pi((a^2 + b^2)/2)^(1/2) and
that of Muir (1883): 2pi((a^(3/2) + b^(3/2))/2)^(2/3).
All three of these approximations can be written in the form
c(a^p + b^p)^(1/p), where c and p are constants. In terms of the
exponents, the YNOT formula is intermediate between the Euler and Muir
formulae because 3/2 < ln(2)/ln(pi/2) < 2. The two older formulae are
not perfect for e=1, although, like all other well known formulae, they
are perfect for e=0.
***Now for something new***
The YNOT formula can be rewritten as 4a(1 + (b/a)^y))^(1/y). For an
approximation of this type to be extreme-perfect, the outer exponent
must be ln(pi/2)/ln(2) or equivalently lg(pi)-1, where lg denotes the
binary (i.e. base 2) logarithm. However, extreme-perfection does not
depend on the inner exponent and, as such, that exponent can be
adjusted in an attempt to improve the approximation. The inner exponent
in the YNOT formula is ln(2)/ln(pi/2) or approx. 1.5349. If this is
reduced somewhat, the revised formula will then underestimate the
perimeter for smaller eccentricities, overestimate for larger
eccentricities, and of course be exact for some intermediate
eccentricity. At which intermediate eccentricity do we wish this to
occur? When e=1/2? Or perhaps when b/a=1/2? The answer was not
immediately obvious to me. In fact, it happens to work beautifully if,
instead of considering either e or b/a directly, we consider the
quantity (a-b)/(a+b) and choose the inner exponent so that
(a-b)/(a+b)=1/2. This condition is equivalent to requiring b/a=1/3
or e=2*Sqrt(2)/3. With a little work, it can be shown that the inner
exponent r must then be
r = - log_3 ( (E(8/9))^(1/(lg(pi)-1)) -1 ) or approx. 1.56186
where log_3 is the ternary (i.e. base 3) logarithm and E is the
complete elliptic integral of the second kind. The new formula is then
perimeter is approx. 4a(1+ (b/a)^r)^(lg(pi)-1).
The maximum of |relative error| is less than 0.00183, which represents
a reduction by a factor of almost 2 compared with the YNOT formula.
***Is anyone aware of other nice extreme-perfect approximation formulae
for the perimeter of an ellipse?
***By the way, I don't particularly like (or dislike) "extreme-
perfect", but couldn't think of a better name. Would anyone care to
suggest another?
***Finally, while we're talking about ellipses and arc lengths, but now
ignoring extreme-perfection: Is anyone aware of a nice formula for
approximating general partial arc lengths, say, approximating the arc
length along an ellipse, parameterized as x=a*cos(t), y=b*sin(t), just
from t=alpha to t=beta?
David W. Cantrell
DWCan...@sigmaxi.org
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Kaimbridge
David W. Cantrell <DWCan...@sigmaxi.org> wrote:
<snip>
> ***Now for something new***
> The YNOT formula can be rewritten as 4a(1 + (b/a)^y))^(1/y).
>
> ***Is anyone aware of other nice extreme-perfect approximation
> formulae for the perimeter of an ellipse?
>
> ***By the way, I don't particularly like (or dislike) "extreme-
> perfect", but couldn't think of a better name. Would anyone care
> to suggest another?
See my post from long ago on "approxallation":
http://forum.swarthmore.edu/epigone/sci.math/chersmexshand
(note: "Za" = Iota)
~Kaimbridge~
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Global 2000 Spheroid [G2KS]: a = 6378.135, b = 6356.75, Gr = 6372.7994