Great-circle radius of ellipsoid

David W. Cantrell wrote in message
...
wrote:
Is there a equation for the thearetical great-circle radius
of a ellipsoid? I understand that a ellipsoid won't have
a single radius but is there any "best-fit" spherical
avarage known?

There may be several known. According to precisely which criterion
do you want the "best-fit"?

A similar but simpler question might be asked about an ellipse,
having semiaxes lengths a and b. Two "average radii" include

the arithmetic mean, (a + b)/2

and

the root-mean-square, Sqrt((a^2 + b^2)/2).

Not quite, those are *approximations* for both the "Gaussian
Mean Radius" and "Rectified Radius" (i.e., north-south, meridional
arcradius).
The approximation--if not THE true equation--for the "best fit",
"spherical great circle" mean arcradius would be:

1 a^2 + b^2
Gr = [- * (a^2 + ---------)]^.5 = [.25 * (3*a^2 + b^2)]^.5
2 2

I.e., the equator, itself, which equals "a", is half of the average!
See my main reply to this thread.

BTW, are *ALL* "ellipsoids of revolution" spheroids, or just ones
that are "nearly spherical"?. E.g., if a = 10000 and b = 2000,
would that be considered a spheroid? If not, what is the minimum
ratio required?

~Kaimbridge~

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(Kaimbridge) wrote:
David W. Cantrell wrote in message
...
wrote:
Is there a equation for the thearetical great-circle radius
of a ellipsoid? I understand that a ellipsoid won't have
a single radius but is there any "best-fit" spherical
avarage known?

There may be several known. According to precisely which criterion
do you want the "best-fit"?

A similar but simpler question might be asked about an ellipse,
having semiaxes lengths a and b. Two "average radii" include

the arithmetic mean, (a + b)/2

and

the root-mean-square, Sqrt((a^2 + b^2)/2).

Not quite,

No, quite. I merely said that they were two "average radii". They are.
I made no claims that they were equal to some quantities which you seem
to have in mind.

those are *approximations* for both the "Gaussian Mean Radius" and
"Rectified Radius" (i.e., north-south, meridional arcradius).

You seem to be talking about a (3-D) spheroid now. I was, as I had clearly
specified, just talking about an ellipse.

The approximation--if not THE true equation--for the "best fit",
"spherical great circle" mean arcradius would be:

1 a^2 + b^2
Gr = [- * (a^2 + ---------)]^.5 = [.25 * (3*a^2 + b^2)]^.5
2 2

I.e., the equator, itself, which equals "a", is half of the average!
See my main reply to this thread.

Thanks, but I think I'll pass.

BTW, are *ALL* "ellipsoids of revolution" spheroids, or just ones
that are "nearly spherical"?.

All. But that's merely a matter of terminology.

DWC

David W. Cantrell wrote in message ...
(Kaimbridge) wrote:
David W. Cantrell wrote in message
...
wrote:
Is there a equation for the thearetical great-circle radius
of a ellipsoid? I understand that a ellipsoid won't have
a single radius but is there any "best-fit" spherical
avarage known?

There may be several known. According to precisely which criterion
do you want the "best-fit"?

A similar but simpler question might be asked about an ellipse,
having semiaxes lengths a and b. Two "average radii" include

the arithmetic mean, (a + b)/2

and

the root-mean-square, Sqrt((a^2 + b^2)/2).

Not quite,

No, quite. I merely said that they were two "average radii". They are.
I made no claims that they were equal to some quantities which you seem
to have in mind.

those are *approximations* for both the "Gaussian Mean Radius" and
"Rectified Radius" (i.e., north-south, meridional arcradius).

You seem to be talking about a (3-D) spheroid now. I was, as I had clearly
specified, just talking about an ellipse.

Right, okay, but even for an ellipse, the "average/mean radii" of that
ellipse equals "a" times the divided difference of the complete elliptic
integral of the second kind, wouldn't you agree?
Yes, .5 * [a + b] and [.5 * (a^2 + b^2)]^.5 are averages of "a" and "b",
but not of the WHOLE ELLIPSE--and the OP seems interested in the whole great
ellipse average.
So, as I said, it would seem,

The approximation--if not THE true equation--for the "best fit",
"spherical great circle" mean arcradius would be:

1 a^2 + b^2
Gr = [- * (a^2 + ---------)]^.5 = [.25 * (3*a^2 + b^2)]^.5
2 2

I.e., the equator, itself, which equals "a", is half of the average!

~Kaimbridge~

-----
Wanted—Kaimbridge (w/mugshot!):
http://www.angelfire.com/ma2/digitol...nted_KMGC.html
----------
Digitology—The Grand Theory Of The Universe:
http://www.angelfire.com/ma2/digitology/index.html

***** Void Where Permitted; Limit 0 Per Customer. *****

(Kaimbridge) wrote:
David W. Cantrell wrote in message
...
(Kaimbridge) wrote:
David W. Cantrell wrote in message
...
wrote:
Is there a equation for the thearetical great-circle radius
of a ellipsoid? I understand that a ellipsoid won't have
a single radius but is there any "best-fit" spherical
avarage known?

There may be several known. According to precisely which criterion
do you want the "best-fit"?

A similar but simpler question might be asked about an ellipse,
having semiaxes lengths a and b. Two "average radii" include

the arithmetic mean, (a + b)/2

and

the root-mean-square, Sqrt((a^2 + b^2)/2).

Not quite,

No, quite. I merely said that they were two "average radii". They are.
I made no claims that they were equal to some quantities which you seem
to have in mind.

those are *approximations* for both the "Gaussian Mean Radius" and
"Rectified Radius" (i.e., north-south, meridional arcradius).

You seem to be talking about a (3-D) spheroid now. I was, as I had
clearly specified, just talking about an ellipse.

Right, okay, but even for an ellipse, the "average/mean radii" of that
ellipse equals "a" times the divided difference of the complete elliptic
integral of the second kind, wouldn't you agree?

No. There is no unique thing which should be called _the_ mean radius of
an ellipse. There are many different means. Yes, there is _a_ mean radius
which would be defined in terms of a complete elliptic integral of the
second kind.

DWC