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From: Kaimbridge on 20 Jul 2010 21:19
For an ellipse/oblatum, first eccentricity^2, e^2, equals
(a^2-b^2)/a^2, while second e^2, e'^2, equals (a^2-b^2)/b^2.
There is also a third e^2, e''^2 or m, equalling (a^2-b^2)/(a^2+b^2).
Relatedly, first flattening, f, equals (a-b)/a, while second f, f' or
n, equals (a-b)/(a+b).
Is there a prescribed distinction between 1st and 2nd orders(?) or
are their designations arbitrary (e.g., could (a^2-b^2)/b^2 just as
rightfully be first e^2?)?
And is there an e''', e'''', etc., and f'', f''', etc.?
The reason I ask, is because if they are expressed as functions of
angular eccentricity, oe, where oe = arccos(b/a), then

a^2 - b^2
e^2 = sin(oe)^2 = ---------;
a^2

sin(oe)^2 sin(oe)^2 a^2 - b^2
e'^2 = tan(oe)^2 = ------------- = ------------- = -----------;
1 - sin(oe)^2 0 + cos(oe)^2 0*a^2 + b^2

sin(oe)^2 sin(oe)^2 a^2 - b^2
e''^2 = ------------- = ------------- = -----------;
2 - sin(oe)^2 1 + cos(oe)^2 1*a^2 + b^2

Continuing on with the same process—?:

sin(oe)^2 sin(oe)^2 a^2 - b^2
?? e'''^2 = ------------- = ------------- = -----------; ??
3 - sin(oe)^2 2 + cos(oe)^2 2*a^2 + b^2

sin(oe)^2 sin(oe)^2 a^2 - b^2
?? e''''^2 = ------------- = ------------- = -----------; ??
4 - sin(oe)^2 3 + cos(oe)^2 3*a^2 + b^2

e...^2 = ... etc. ??

The same approach can be applied to flattening:

a - b
f = 2*sin(.5*oe)^2 = 1 - cos(oe) = -----;
a

2*sin(.5*oe)^2 1 - cos(oe) a - b
f' = tan(.5*oe)^2 = -------------------- = ----------- =
-------;
2*(1 - sin(.5*oe)^2) 1 + cos(oe) 1*a + b

2*sin(.5*oe)^2 1 - cos(oe) a - b
?? f'' = -------------------- = ----------- =
-------; ??
2*(2 - sin(.5*oe)^2) 3 + cos(oe) 3*a + b

2*sin(.5*oe)^2 1 - cos(oe) a - b
?? f''' = -------------------- = ----------- =
-------; ??
2*(3 - sin(.5*oe)^2) 5 + cos(oe) 5*a + b

f... = ... etc.

Is this the right extension process (if there is one)?

And what about a prolatum, where the "a"s and "b"s are swapped for all
but the denominator of f, which remains "a", resulting in a secant
complement?

oe_p = acos(a/b);

b^2 - a^2
e^2 = sin(oe_p)^2 = ---------;
b^2

sin(oe_p)^2 sin(oe_p)^2 b^2 - a^2
e'^2 = tan(oe_p)^2 = --------------- = --------------- =
-----------;
1 - sin(oe_p)^2 0 + cos(oe_p)^2 0*b^2 + a^2

sin(oe_p)^2 sin(oe_p)^2 b^2 - a^2
e''^2 = --------------- = --------------- =
-----------;
2 - sin(oe_p)^2 1 + cos(oe_p)^2 1*b^2 + a^2

e...^2 = ... etc. ??

and
b - a
f = sec(oe_p) - 1 = -----;
a

sec(oe_p) - 1 1 - cos(oe_p) b - a
f' = tan(.5*oe_p)^2 = ------------- = ------------- =
-------;
sec(oe_p) + 1 1 + cos(oe_p) 1*b +
a

sec(oe_p) - 1 1 - cos(oe_p) b - a
?? f'' = --------------- = ------------- =
-------; ??
3*sec(oe_p) + 1 3 + cos(oe_p) 3*b +
a
or
sec(oe_p) - 1 1 - cos(oe_p) b - a
? = ------------- = --------------- =
-------; ??
sec(oe_p) + 3 1 + 3*cos(oe_p) b +
3*a

f... = ... etc.

If all of these aren't right, is there a formula for calculating a
given eccentricity or flattening order?

~Kaimbridge~

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From: Henry on 21 Jul 2010 04:56
On 21 July, 02:19, Kaimbridge <kaimbri...(a)gmail.com> wrote:
> For an ellipse/oblatum, first eccentricity^2, e^2, equals
> (a^2-b^2)/a^2, while second e^2, e'^2, equals (a^2-b^2)/b^2.
> There is also a third e^2, e''^2 or m, equalling (a^2-b^2)/(a^2+b^2).

So http://en.wikipedia.org/wiki/Eccentricity_(mathematics) says

> Relatedly, first flattening, f, equals (a-b)/a, while second f, f' or
> n, equals (a-b)/(a+b).
> Is there a prescribed distinction between 1st and 2nd orders(?) or
> are their designations arbitrary (e.g., could (a^2-b^2)/b^2 just as
> rightfully be first e^2?)?

There is a distinction in that a >= b, and this means your first
eccentricity of an ellipse corresponds the classical eccenticity of a
conic section: the ratio between the distance to the focus and the
distance to the directrix.

> And is there an e''', e'''', etc., and f'', f''', etc.?

If you define them, then perhaps. But they would not be standard and
you would have to show they had interesting properties if you wanted
others to consider them.

From: Kaimbridge on 21 Jul 2010 14:35
On Jul 21, 8:56 am, Henry <s...(a)btinternet.com> wrote:
> On 21 July, 02:19, Kaimbridge <kaimbri...(a)gmail.com> wrote:

>> Relatedly, first flattening, f, equals (a-b)/a, while second f,
>> f' or n, equals (a-b)/(a+b).
>> Is there a prescribed distinction between 1st and 2nd orders(?)
>> or are their designations arbitrary (e.g., could (a^2-b^2)/b^2
>> just as rightfully be first e^2?)?
>
> There is a distinction in that a >= b, and this means your first
> eccentricity of an ellipse corresponds the classical eccenticity
> of a conic section: the ratio between the distance to the focus
> and the distance to the directrix.
>
>> And is there an e''', e'''', etc., and f'', f''', etc.?
>
> If you define them, then perhaps. But they would not be
> standard and you would have to show they had interesting
> properties if you wanted others to consider them.

Well, the idea was if there was an inherent formula structure to
a progressive succession, that could be used someway, such as some
sort of series: E.g., V = (e*Q)^2 + (e'*Q)^4 + (e''*Q)^6 + ....

But, okay, let's limit it to the known orders/degrees.
For an ellipse and oblatum,

L = (a^2 - b^2)^.5 = Linear eccentricity;
L'= (a^2 + b^2)^.5 = ???

e = L/a; e' = L/b; e'' = L/L'.

What is L'?
Similarly,

a - b a^2 - a*b
f = ----- = --------- = 2*sin(.5*oe)^2;
a a^2

a - b a^2 - a*b
f' = ----- = --------- = tan(.5*oe)^2;
a + b a^2 + a*b

Viewing f this way, one could define "regular" eccentricity as
"plane eccentricity" and flattening as "surface eccentricity^2",
f = g^2, where a^2 is the equatorial geometric mean radius^2
(triaxially, a_x*a_y) and a*b is the meridional geometric mean
radius^2. Continuing this approach,

S = (a^2 - a*b)^.5 = Surface eccentricity (?);
S'= (a^2 + a*b)^.5 = ???

g = f^.5 = S/a; g' = (f')^.5 = S/S'.

Since the ratios involve meridian-equator, there is no S/b,
and for a prolatum g remains S/a, though the positions of a^2
and a*b in S and S' are reversed.
Like L', what is S'?

~Kaimbridge~

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