Historical Oblate Spheroid Parameters Calculator
Given the Geographic Coordinates and Elevation of Any Point Relative
to the Surface of Any of Several Historical Reference Spheroid Models
Latitude  (−Neg = South)   ,  Longitude  (−Neg = West)
±Deg.dddddddddd   or   ±Deg  min  sec.ssss
Sea-Level Elevation 
in meters or feet (m,   ft)
Select a spheroid model to compute the rectangular 3D  XYZ-coordinates of the point based
on the geographic coordinates and elevation given above and the equations following below.


Given the general static ellipsoid parameters:
  • R = Equatorial radius of ellipsoid = Length of semi-major axis
  • r = Polar radius of ellipsoid = Length of semi-minor axis
  • where,  rR

  • If ((R = r)), then the form is a perfect sphere, otherwise it is an ellipsoid or oblate
    spheroid to some degree, generally wider between two opposite equatorial points
    due to a slight equatorial bulge, than between the opposite polar points.

  • NOTE:
    In the equations, any units of measure may be used for (R, r), such as feet, meters,
    kilometers, miles, etc., just as long as the same units are used consistently throughout.
    The values of (x, y, z) will be expressed in the same units as the parameters (R, r).

Computing 3D Rectangular XYZ-Coordinates For a Point on a Reference Ellipsoid Surface

The reference ellipsoid parameters, location and elevation of the point are:
R = Equatorial radius.
r = Polar radius.

Lat = Geographic latitude of point on ellipsoid surface.
Lon = Geographic longtitude of point on ellipsoid surface (Negative = West).

h = Height or elevation of point relative to ellipsoid surface.

From the given parameters, the 3D XYZ-coordinates of the point, relative to the surface
of the reference ellipsoid, are computed from the equations given below.


f = R     r R = Polar flattening factor of ellipsoid.

e = 1 ( r R ) 2

ρ = R 1     e ² · s i n ² ( Lat ) = Radius vector from ellipsoid center to surface at latitude (Lat).


x = ( h + ρ ) · c o s ( Lat ) · c o s ( Lon )

y = ( h + ρ ) · c o s ( Lat ) · s i n ( Lon )

z = ( h + ρ · ( 1 e ² ) ) · s i n ( Lat )


The Earth Ellipsoid

A PHP Science Program by Jay Tanner - Revised: 2023 March 01 Wednesday at 11:53:49 PM GMT