Two types of cartesian coordinates?

[Nomen Nescio]

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'?

TIAWL

[William Elliot]

On Wed, 13 Jun 2012, Nomen Nescio wrote:

> I have seen two different types of cartesian coordinates:
>
> x = a * cos(theta)
> y = b * sin(theta)

Those ore parametric equations for the ellipse
x^2 / a^2 + y^2 / b^2 = 1

> and
> x' = a * sin(theta)
> y' = b * cos(theta)

Those ore parametric equations for the ellipse
x'^2 / a^2 + y'^2 / b^2 = 1

> or
> x' = b * cos(theta)
> y' = a * sin(theta)
>

Those ore parametric equations for the ellipse
x'^2 / b^2 + y'^2 / a^2 = 1

> What is the difference between x,y and x',y', in terms of names (identities?)
> and their basic meanings?

You defined x' and y' in two different ways.

> I believe x and y are known as the "parametric equation of ellipse", but
> what about x' and y'?

Parametric equation of an ellipse with the parameter theta. x' and y' are
just two additional variables that can be used, if desired, to name the
horizontal and vertical axis.

[Aielyn]

Your first two equations are identical up to a linear transformation of theta.

That is, if you let theta' = pi/2 - theta, then one can write the first equations as

x = a * sin (theta')
y = b * cos (theta')

Which is identical to your second equations, except with x and y replaced by x' and y', and theta' replaced by theta. The third equations are, of course, just switching x' and y' around, and therefore produce an ellipse oriented vertically instead of horizontally (assuming |a|>|b|).

[KBH]

In land surveying where North is vertical or zero-azimuth is straight
up then:

y = distance * Cos(azimuth)
x = distance * Sin(azimuth)

[Kaimbridge]

[ For coherent viewing, fixed-width font
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character encoding should be utilized ]

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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[Kaimbridge]

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.math/msg/990fce2fabd6e407?dmode=source&output=gplain

[Kaimbridge]

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;

[Frederick Williams]

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.math/msg/990fce2fabd6e407?dmode=source&output=gplain

Alternatively, you could write in ASCII.

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