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Sep 29, 2007, 3:34:14 PM9/29/07

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On Jul 11, 3:33 pm, Narasimham <mathm...@hotmail.com> wrote:

> On Jul 11, 5:42 pm, George Orwell <nob...@mixmaster.it> wrote:

> ----

>

>> Ellipsoid:

>> x=a*sin(theta)*cos(phi),

>> y=b*sin(theta)*sin(phi),

>> z=c*cos(theta).

>

>> Is it

>

>> x'=b*c*sin(theta)*cos(phi),

>> y'=a*c*sin(theta)*sin(phi),

>> z'=a*b*cos(theta)?

Yes, if "theta" equals the azimuth, using the spherical coordinate

system.

>> or

>> x'=b*c*cos(theta)*cos(phi),

>> y'=a*c*cos(theta)*sin(phi),

>> z'=a*b*sin(theta)?

Yes, if "theta" equals the *parametric* (or "reduced") latitude,

using

the geographic coordinate system.

> ---

> If you continue to use ' latitude and longitude" notation, these

> are same ellipsoids but with axes interchanged:

>

>> x'=bc*cos(theta)*cos(phi),

>> y'=ac*cos(theta)*sin(phi),

>> z'=ab*sin(theta)? -> lat,long = theta, phi

Right, but--again--it's the parametric "RLat" (usually denoted as

\beta), not the more recognized geographic "Lat" (usually denoted as

\phi).

There are a couple of different issues involved here.

--What are x', y', z'--derivatives or complements?

--While (AFAIK) it hasn't been recognized as such,

a scalene ellipsoid *CAN* be defined biaxially,

like a spheroid.

First, let's look at an oblate spheroid, which can be defined either

biaxially, like an ellipse, or triaxially:

Where RL = reduced/parametric latitude;

Lon = geographic longitude;

a, b = equatorial, polar radii;

Biaxial: x = a * cos(RL);

y = b * sin(RL);

Triaxial: X = a * cos(RL) * cos(Lon);

Y = a * cos(RL) * sin(Lon);

Z = b * sin(RL);

x = x(Lon) = [X^2 + Y^2]^.5,

= cos(RL) * [(a*cos(Lon))^2 + (a*sin(Lon))^2]^.5,

= cos(RL) * a(Lon);

R = [x^2 + y^2]^.5 = [X^2 + Y^2 + Z^2]^.5,

= [(a*cos(RL))^2 + (b*sin(RL))^2]^.5;

Now let's consider the complementary/auxiliary parameterization, first

by redefining the triaxial "a", "b", "c" radii:

Equatorial: a_x = "a"; a_y = "b";

Polar: b = "c";

Furthermore, the geometric mean of a_x and a_y, "a_m", the variable

equatorial radius, "a(Lon)", and its complement, "b(Lon)", will now be

introduced:

(Note: Read V' as "complement V", where the

acute accent, "'", is over the V)

a_m = [a_x * a_y]^.5;

a(Lon) = [(a_x*cos(Lon))^2 + (a_y*sin(Lon))^2]^.5;

a_y * b a_y

a_x'= ------- = b * [---]^.5;

a_m a_x

a_x * b a_x

a_y'= ------- = b * [---]^.5;

a_m a_y

a'(Lon) = b(Lon) = [(a_x'*cos(Lon))^2 + (a_y'*sin(Lon))^2]^.5,

a(90°+/-Lon)

= b * ------------,

a_m

a_y a_x

= b * [---*cos(Lon)^2 + ---*sin(Lon))^2]^.5;

a_x a_y

Thus, for an oblate spheroid,

a_x = a_y = a; a_x' = a_y' = a' = b;

The spheroid's complementary/auxiliary parameterization can now be

defined:

Biaxial: x'= a * cos'(RL) = a * -sin(RL);

y'= b * sin'(RL) = b * cos(RL);

or

x'= a' * cos(RL) = b * cos(RL);

y'= b' * sin(RL) = a * sin(RL);

Triaxial: X'= a * cos'(RL) * cos(Lon) = a * -sin(RL) * cos(Lon);

Y'= a * cos'(RL) * sin(Lon) = a * -sin(RL) * sin(Lon);

Z'= b * sin'(RL) = b * cos(RL);

or

X'= a' * cos(RL) * cos(Lon) = b * cos(RL) * cos(Lon);

Y'= a' * cos(RL) * sin(Lon) = b * cos(RL) * sin(Lon);

Z'= b' * sin(RL) = a * sin(RL);

Since the functions of Lon in X' and Y' are cofactors of sin(RL) (and

I believe they should still be cofactors of cos(RL)), it would seem

that the valid complementary parameterization is based on a' and b',

rather than cos'(RL) and -sin'(RL).

Thus,

R'= [x'^2 + y'^2]^.5 = [X'^2 + Y'^2 + Z'^2]^.5,

= [(b*cos(RL))^2 + (a*sin(RL))^2]^.5;

With a scalene ellipsoid, the same exact formulation as the oblate

spheroid applies to the primary parameterization, only substituting a

with a(Lon):

Triaxial: X = a_x * cos(RL) * cos(Lon);

Y = a_y * cos(RL) * sin(Lon);

Z = b * sin(RL);

Biaxial: x(Lon) = a(Lon) * cos(RL) = [X^2 + Y^2]^.5;

y = b * sin(RL) = Z;

R = [x(Lon)^2 + y^2]^.5 = [X^2 + Y^2 + Z^2]^.5,

= [(a(Lon)*cos(RL))^2 + (b*sin(RL))^2]^.5;

The presentation of the complementary/auxiliary parameterization

likewise mirrors that of the oblate spheroid's, with one important

caveat: Generally, the scalene's complementary parameterization is

presented in the context of its surface area integrand:

(RS(RL)^2)' = a_m * cos(RL) * [X'^2 + Y'^2 + Z'^2]^.5

Thus, the OP and follow-up both erroneously factored in a_m with

X', Y', Z'. The proper assignment works out as

X'= a_x' * cos(RL) * cos(Lon),

a_y * b

= b(0) * cos(RL) * cos(Lon) = ------- * cos(RL) * cos(Lon);

a_m

Y'= a_y' * cos(RL) * sin(Lon),

a_x * b

= b(90°) * cos(RL) * sin(Lon) = ------- * cos(RL) * sin(Lon);

a_m

a_x * a_y

Z'= b' * sin(RL) = a_m * sin(RL) = --------- * sin(RL);

a_m

x'(Lon) = a'(Lon) * cos(RL) = b(Lon) * cos(RL) = [X^2 + Y^2]^.5;

y'= b' * sin(RL) = a_m * sin(RL) = Z;

R'= [x'(Lon)^2 + y^2]^.5 = [X'^2 + Y'^2 + Z'^2]^.5,

= [(b(Lon)*cos(RL))^2 + (a_m*sin(RL))^2]^.5;;

A reply to another thread ("Principle perimetres of triaxal

ellipsoid") is being finished up, which apply these concepts.

~Kaimbridge~

-----

Wikipedia-Contributor Home Page:

http://en.wikipedia.org/wiki/User:Kaimbridge

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

</pre>

Oct 1, 2007, 7:34:59 PM10/1/07

to

Right, but——again——it's the parametric "RLat" (usually denoted as

\beta), not the more recognized geographic "Lat" (usually denoted as \phi).

There are a couple of different issues involved here.

--What are x', y', z'——derivatives or complements?

~Kaimbridge~

-----

Wikipedia—Contributor Home Page:

Oct 1, 2007, 7:57:37 PM10/1/07

to

On Sep 29, 7:34 pm, Kaimbridge <Kaimbri...@gmail.com> wrote:

> <pre>

> [ For coherent viewing, fixed-width font (such as

> "courier new") and "UTF-8" character encoding

> should be utilized ]

<snip>> <pre>

> [ For coherent viewing, fixed-width font (such as

> "courier new") and "UTF-8" character encoding

> should be utilized ]

> (Note: Read V' as "complement V", where the

> acute accent, "'", is over the V)

> </pre>

http://groups.google.com/group/sci.math/msg/8733012fc614ca70?dmode=source&output=gplain

Reposted for notation clarification (the original Google posting

changed V' to just V'--thus muddying the distinction) and better

propagation.

Oct 6, 2007, 6:51:01 PM10/6/07

to

On Oct 1, 11:34 pm, "Kaimbridge M. GoldChild" <Kaimbri...@Gmail.com>

wrote:

wrote:

> There are a couple of different issues involved here.

>

> --What are x', y', z'--derivatives or complements?

>

> --While (AFAIK) it hasn't been recognized as such,

> a scalene ellipsoid *CAN* be defined biaxially,

> like a spheroid.

<snip>

> A reply to another thread ("Principle perimetres of triaxal ellipsoid")

> is being finished up, which apply these concepts.

Just posted:

http://groups.google.com/group/sci.math/msg/22b36d181b87df41

http://groups.google.com/group/sci.math/msg/22b36d181b87df41?output=gplain

> ~Kaimbridge~

>

> -----

> Wikipedia-Contributor Home Page:

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