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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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