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Two types of cartesian coordinates? (*** Formatting Fixed ***)



 
 
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  #1  
Old June 14th 12, 09:52 PM posted to sci.math,sci.physics,sci.astro
Kaimbridge
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Default Two types of cartesian coordinates? (*** Formatting Fixed ***)

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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

λ is the geographical/geodetic longitude;

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

then

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

and

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

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

and

x΄ = b * cos(β); y΄= a * sin(β);

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

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

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

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

__ β_f __ φ_f
/ /
DxM = / S(β)dβ = / M(φ)dφ;
__/ __/
β_s φ_s

and

__ β_f
Surface /
Area = Δλ a / cos(β)*S(β)dβ,
__/
β_s

__β_f __ λ_f
/ /
= a_m / cos(β) / (x΄(λ)^2 + y΄^2)^.5 dλdβ
__/ __/
β_s λ_s

~Kaimbridge~

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  #2  
Old June 14th 12, 09:58 PM posted to sci.math,sci.physics,sci.astro
Kaimbridge
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Default Two types of cartesian coordinates? (*** Formatting Fixed ***)

On 12.Jun.14.Thu 20:52 (UTC), Kaimbridge wrote:

[ For coherent viewing, fixed-width font
(such as “courier new”) and “UTF-8”
character encoding should be utilized ]


If posting still doesn't render properly, try this link:


https://groups.google.com/group/sci....&output=gplain

~Kaimbridge~

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  #3  
Old June 14th 12, 10:09 PM posted to sci.math,sci.physics,sci.astro
Kaimbridge
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Posts: 8
Default Two types of cartesian coordinates? (*** Formatting Fixed ***)

On 12.Jun.14.Thu 20:52, Kaimbridge wrote:

Where

λ is the geographical/geodetic longitude;

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

^^^ ^^^

That, obviously, should be:

a(λ) = ((a_x*cos(λ))^2 + (a_y*sin(λ))^2)^.5;

~Kaimbridge~

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  #4  
Old June 14th 12, 11:05 PM posted to sci.math,sci.physics,sci.astro
Frederick Williams[_3_]
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Default Two types of cartesian coordinates? (*** Formatting Fixed ***)

Kaimbridge wrote:

On 12.Jun.14.Thu 20:52 (UTC), Kaimbridge wrote:

[ For coherent viewing, fixed-width font
(such as “courier new”) and “UTF-8”
character encoding should be utilized ]


If posting still doesn't render properly, try this link:


https://groups.google.com/group/sci....&output=gplain


Alternatively, you could write in ASCII.

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