A new approximation for elliptic arc lengths

pre

David W. Cantrell wrote:
("Clem T.") wrote:
# On 28 Dec 2004, David W. Cantrell wrote:
# A small correction is mentioned below. Better late than never,
# I hope.
#
# David W. Cantrell wrote:
% For the perimeter of an ellipse, many approximations have been
% proposed, some of which are well known. But unfortunately, for
% lengths of general elliptic arcs, such cannot be said.
#
# What about Andoyer's approximation?:

Yes!

# http://www.voidware.com/earthdist.htm
#
# /* approx distance between points on earth ellipsoid.
# * H.Andoyer from Atronomical Algorithms, Jean Meeus, second
# * edition.
# */
# double f = (lat1 + lat2)/2*RAD;
# double g = (lat1 - lat2)/2*RAD;
# double l = (long1 - long2)/2*RAD;
#
# double sg = sin(g);
# double sl = sin(l);
# double sf = sin(f);
#
# double s, c, w, r, d, h1, h2;
# const double a = 6378.14; /* equator earth radius */
# const double fl = 1/289.257; /* earth flattening */

we should have fl = 1/298.257 instead.

Right, as well as "Atronomical" should be "Astronomical"! P=)

# sg = sg*sg;
# sl = sl*sl;
# sf = sf*sf;
#
# s = sg*(1-sl)+(1-sf)*sl;
# c = (1-sg)*(1-sl)+sf*sl;
#
# w = atan(sqrt(s/c));
# r = sqrt(s*c)/w;
# d = 2*w*a;
# h1 = (3*r-1)/2/c;
# h2 = (3*r+1)/2/s;
#
# return d*(1+fl*(h1*sf*(1-sg)-h2*(1-sf)*sg))

Thanks for mentioning Andoyer's approximation. I had not seen it
before.

As given by Meeus (I've seen several sources of his particular
presentation), even excluding the obvious algorithm formatting,
the presented formulation is needlessly complicated--try this:

ADg = standard (graticular) angular distance, found via
spherical "cosine for sides" equation (as given above,
.5 * ADg is found);

F{Q_1}^2 = csc{ADg}^2 * [
3 * (.5 * sin{Lat1}^2 + sin{Lat2}^2 * sinc{2*ADg}
- sin{Lat1} * sin{Lat2} * sinc{ADg})
+ .5 * (sin{Lat1}^2 + sin{Lat2}^2)
- sin{Lat1} * sin{Lat2} * cos{ADg} ];

EGaDD = 1 - f * F{Q_1}^2;

Distance = a * EGDD * ADg,
~=~ a * EGaDD * ADg;

It may be perfectly acceptable for approximating arc lengths on
the Earth's surface. But the Earth's eccentricity is small
(e 0.1) and it's relatively easy to get approximations which
work well specifically when eccentricity is small.

My simple approximation provides |relative error| 0.56%,
regardless of eccentricity. By comparison, if my calculations are
correct, |relative error| for Andoyer's approximation can exceed
21% when eccentricity is large.

Not so quick! P=)
You are considering the traditionally recognized, CONFORMAL arc
length (which involves parametric/reduced latitudes and
parametric/conformal angular distance--"RLat" and "ADc"), but this
approximation uses only the graticular Lat and ADg; thus this
formulation is approximating the GRATICULAR arc length.
See my recent posts on the concept:


http://groups.google.ca/groups?selm=...Programmer.Net

http://groups.google.ca/groups?selm=...Programmer.Net

If you find the distance between the equator and the pole, a * EGaDD
equals .5 * [a + b]. Likewise, if you find the distance from the
equator out to the transverse equator, along the 45° arc path (i.e.,
Lat1 = Long1 = 0, Lat2 = 45°, Long2 = 90°), a * EGaDD equals
..25 * [(3 *a) + b].
As you are well aware, ".5 * [a + b]" is less than the true value of
the mean arcradius of an ellipse, and "[.5 * (a^2 + b^2)]^.5" is
nearly equally larger, with Muir's "[.5 * (a^1.5 + b^1.5)]^(1/1.5)"
nearly perfect for Earth!
So, if there was a way to change Andoyer's EGDD approximation from
a ".5 * [a + b]" factor to "[.5 * (a^1.5 + b^1.5)]^(1/1.5)", results
may be greatly improved--but, again, what will be found is the
graticular arc length, not the conformal--though, for Earth, in most
cases there isn't much of a difference!
But, a more important question to ask first, is whether Andoyer's
approximation is, in fact, an approximation, or the first term of
some (binomial?) series expansion--?

Let cos{Oz} = b/a, f = 1 - cos{Oz} = 2 * sin{.5*Oz}^2:

EGaDD = 1 - f * F{Q_1}^2,
= 1 - 2 * sin{.5*Oz}^2 * F{Q_1}^2;

Could it expand out?

= 1 - CF_1 * sin{.5*Oz}^2 * F{Q_1}^2
+ CF_2 * sin{.5*Oz}^4 * F{Q_2}^4
- CF_3 * sin{.5*Oz}^6 * F{Q_3}^6 + ...

where CF_tn is the term's coefficient and F{Q_tn}^(2*TN) is an
integrated function, not interchangeable (e.g., F{Q_2}^4 x=x
[F{Q_1}^2]^2), in the same way that

__UB __UB
/ (sin{P}^2)dP = [ 1 - / cos{2*P}dP ]/2^1;
__/ __/
LB LB

and

__UB __UB
/ (sin{P}^4)dP = [ 3 - 4 * / cos{2*P}dP
__/ __/
LB LB

__UB
+ / cos{4*P}dP ]/2^3;
__/
LB
where

__UB 1
/ cos{2*TN*P}dP = -- * sin{TN*[UB-LB]} * cos{2*TN*[LB+UB]};
__/ TN
LB

Given its components, could "F{Q_tn}^(2*TN)" be a similar type
situation?

~Kaimbridge~

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"Kaimbridge M. GoldChild" wrote:
David W. Cantrell wrote:
[snip]
Thanks for mentioning Andoyer's approximation. I had not seen it
before.

As given by Meeus (I've seen several sources of his particular
presentation), even excluding the obvious algorithm formatting,
the presented formulation is needlessly complicated--try this:
[snip]
It may be perfectly acceptable for approximating arc lengths on
the Earth's surface. But the Earth's eccentricity is small
(e 0.1) and it's relatively easy to get approximations which
work well specifically when eccentricity is small.

My simple approximation provides |relative error| 0.56%,
regardless of eccentricity. By comparison, if my calculations are
correct, |relative error| for Andoyer's approximation can exceed
21% when eccentricity is large.

Not so quick! P=)
[snip]

If Andoyer's approximation is used to obtain the perimeter of an entire
ellipse (by finding the distance from equator to pole, and then
multiplying that result by 4), we get Kepler's simple approximation

pi*(a + b)

for which |relative error| can exceed 21%, as I noted previously. And so
I can quickly say that such an approximation is of little interest to me.

David W. Cantrell