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Topic: Two types of cartesian coordinates?
Replies: 9   Last Post: Jun 14, 2012 6:05 PM

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Kaimbridge M. GoldChild

Posts: 79
From: 42.57°N/70.89°W; FN42nn (North Shore, Massachusetts, USA)
Registered: 3/28/05
Re: Two types of cartesian coordinates? (*** Formatting Fixed ***)
Posted: Jun 14, 2012 4:52 PM
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On 12.Jun.13.Wed 02:08 (UTC), Nomen Nescio wrote:

> I have seen two different types of cartesian coordinates:
>
> x = a * cos(theta)
> y = b * sin(theta)
> and
> x' = a * sin(theta)
> y' = b * cos(theta)
> or
> x' = b * cos(theta)
> y' = a * sin(theta)
>
> What is the difference between x,y and x',y', in terms of
> names (identities?) and their basic meanings?
> I believe x and y are known as the "parametric equation of
> ellipse", but what about x' and y'?


I don?t know what the formal name is, but basically x' and y'
define the ?parametric equation of the ellipse surface? or,
extending it to three axes??X', Y', Z'??by adding a longitude,
the ?parametric equation of the ellipsoid surface?.

Where

? is the geographical/geodetic longitude;

a_x, a_y are the equatorial radii of their respective axis:
a(?) = ((a*cos(?))^2 + (a*sin(?))^2)^.5;

a_m = b? = (a_x*a_y)^.5;

and

b_x = a_x? = b*(a_y/a_x)^.5 = b*a_y/a_m;
b_y = a_y? = b*(a_x/a_y)^.5 = b*a_x/a_m;
b(?) = a?(?)=((b_x*cos(?))^2 + (b_y*sin(?))^2)^.5;

then

X = a_x * cos(?) * cos(?);
Y = a_y * cos(?) * sin(?);
x(?) = (X^2 + Y^2)^.5 = a(?) * cos(?);
y = Z = b * sin(?);
R(?) = (x(?)^2 + y^2)^.5 = (X^2 + Y^2 + Z^2)^.5;

and

X? = b_x * cos(?) * cos(?);
Y? = b_y * cos(?) * sin(?);
x?(?) = (X?^2 + Y?2)^.5 = b(?) * cos(?);
y? = Z? = a_m * sin(?);
S(?) = R?(?) = (x?(?)^2 + y?^2)^.5,
= (X?^2 + Y?^2 + Z?^2)^.5;

Thus, for an ellipse (and non-scalene spheroid), these reduce to

x = a * cos(?); y = b * sin(?);

R(?) = S(90-?) = (x^2 + y^2)^.5,
= ((a * cos(?))^2 +(b * sin(?))^2)^.5;

and

x? = b * cos(?); y?= a * sin(?);

S(?) = R(90-?) = (x?^2 + y?^2)^.5,
= ((a * sin(?))^2 +(b * cos(?))^2)^.5;

So what does this all mean?
Well, in terms of the surface parameters, rather than derivatives
of ??s trig functions, x? and y? are fundamentally based on radii
complements, as the triaxial case demonstrates.
In terms of uses, R(?) is the integrand for the well known
elliptic integral of the second kind, and S(?) is the auxiliary
integrand for meridional distance, DxM, as well as (authalic)
surface area:

Where ? is the geographical/geodetic latitude and M is the
(conjugate) meridional radius of curvature,

M(?) = (a*b)^2/R(?)^3,
= (a*b)^2/((a * cos(?))^2 +(b * sin(?))^2)^1.5;

__ ?_f __ ?_f
/ /
DxM = / S(?)d? = / M(?)d?;
__/ __/
?_s ?_s

and

__ ?_f
Surface /
Area = ?? a / cos(?)*S(?)d?,
__/
?_s

__?_f __ ?_f
/ /
= a_m / cos(?) / (x?(?)^2 + y?^2)^.5 d?d?
__/ __/
?_s ?_s

~Kaimbridge~

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

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