Heuristic Algorithm to Compute Geodetic Height (h) from Ellipse Equation

The equation of the ellipse in a meridian plan is well known to be which can be defined in its most generic form as where the variable (s) is the indicator value of the ellipsoidal height. For (h=0), the value of  (s) equals (1) , for negative  (h), the value  is lower than (1), and vice versa, for positive (h) , the corresponding  value of (s) is greater than (1). Hence (s) and (h) are highly-correlated. The main goal of this work is to exploit the  (s-h) correlation and to represent it both statistically and mathematically. Moreover, the essential role of (s) in the transformation of the Cartesian coordinates (x_p, y_p, z_p) of any generic point (P) into its geodetic coordinates (φ, λ, h) in reference to the geodetic ellipsoid is thoroughly studied.