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 1:36 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 axesX', 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 nonscalene 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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