An ellipsoid is a mathematically defined surface which approximates the geoid: the surface of the Earth's gravity field, which is approximately the same as mean sea level.
A complete ellipsoid definition comprises a size (primary) and a shape (secondary) parameter.
.. option:: +R=<value> Radius of the sphere, :math:`R`.
.. option:: +a=<value> Semi-major axis of the ellipsoid, :math:`a`.
.. option:: +rf=<value> Reverse flattening of the ellipsoid, :math:`1/f`.
.. option:: +f=<value> Flattening of the ellipsoid, :math:`f`.
.. option:: +es=<value> Eccentricity squared, :math:`e^2`.
.. option:: +e=<value> Eccentricity, :math:`e`.
.. option:: +b=<value> Semi-minor axis, :math:`b`.
The ellipsoid definition may be augmented with a spherification flag, turning the ellipsoid into a sphere with features defined by the ellipsoid.
.. option:: +R_A=<value> A sphere with the same surface area as the ellipsoid.
.. option:: +R_V=<value> A sphere with the same volume as the ellipsoid.
.. option:: +R_a=<value> A sphere with :math:`R = (a + b)/2` (arithmetic mean).
.. option:: +R_g=<value> A sphere with :math:`R = \sqrt{ab}` (geometric mean).
.. option:: +R_h=<value> A sphere with :math:`R = 2ab/(a+b)` (harmonic mean).
.. option:: +R_lat_a=<phi> A sphere with :math:`R` being the arithmetic mean of the corresponding ellipsoid at latitude :math:`\phi`.
.. option:: +R_lat_g=<phi> A sphere with :math:`R` being the geometric mean of the corresponding ellipsoid at latitude :math:`\phi`.
If :option:`+R` is given as size parameter, any shape and spherification parameters given are ignored.
The ellps=xxx
parameter provides both size and shape for a number of
built-in ellipsoid definitions.
ellps Parameters Datum name GRS80 a=6378137.0 rf=298.257222101 GRS 1980(IUGG, 1980) airy a=6377563.396 b=6356256.910 Airy 1830 bessel a=6377397.155 rf=299.1528128 Bessel 1841 clrk66 a=6378206.4 b=6356583.8 Clarke 1866 intl a=6378388.0 rf=297. International 1909 (Hayford) WGS60 a=6378165.0 rf=298.3 WGS 60 WGS66 a=6378145.0 rf=298.25 WGS 66 WGS72 a=6378135.0 rf=298.26 WGS 72 WGS84 a=6378137.0 rf=298.257223563 WGS 84 sphere a=6370997.0 b=6370997.0 Normal Sphere (r=6370997)
If size and shape are given as ellps=xxx
, later shape and size parameters
are are taken into account as modifiers for the built-in ellipsoid definition.
While this may seem strange, it is in accordance with historical PROJ
behavior. It can e.g. be used to define coordinates on the ellipsoid
scaled to unit semimajor axis by specifying +ellps=xxx +a=1
Spherical earth with radius 7000km:
+proj=latlon +R=7000000
Using the GRS80 ellipsoid:
+proj=latlon +ellps=GRS80
Expressing ellipsoid by semi-major axis and reverse flattening (1/f):
+proj=latlon +a=6378137.0 +rf=298.25
Spherical earth based on volume of ellipsoid
+proj=latlon +a=6378137.0 +rf=298.25 +R_V