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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?
Posted: Jun 13, 2012 2:10 PM
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On Jun 13, 2:08 am, 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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