x*x*b*b + y*y*b*b + z*z*a*a = a*a*b*b
|
a: Major semi-axis (O-A), (F-B)
b: Minor semi-axis (O-B) h: Ellipsoidal elevation (P-R) f: Semi-focal length (O-F) f = a*a - b*b = (a + b)*(a - b)
Latitude and longitude: s: Sine latitude c: Cosine latitude p: Sine longitude q: Cosine longitude s = kn
c = SQRT(in*in + jn*jn)
p = jn/c
q = in/c
Normal direction cosines: in = c*q
jn = c*p
kn = s
Prime vertical direction cosines: ip = -s*q
jp = -s*p
kp = c
Geocentric latitude tangent: z/x = (s*b*b)/(c*a*a)
|
t: Tangential plane free term (T-P)
t = -SQRT(a*a*c*c + b*b*s*s)
g: Normal interaxial segment (E-N) g = f/t
o: Prime vertical free term (O-T) o = g*s*c
n: Prime vertical radius (N-P) n = (a*a)/t
l: Normal equatorial depth (E-P) l = (b*b)/t
m: Meridian radius m = (n*l)/t
r: Geometric mean of (n, n) r = (n*b)/t
Coordinates of para-centre (T) u = g*s*s*c*q
v = g*s*s*c*p
w = -q*s*c*c
Coordinates of surface point (P): x = (n + h)*c*q = u - in*t
y = (n + h)*c*p = v - jn*t
z = -q*s*c*c = w - kn*t
|