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diff --git a/docs/source/usage/ellipsoids.rst b/docs/source/usage/ellipsoids.rst new file mode 100644 index 00000000..cd4b0c43 --- /dev/null +++ b/docs/source/usage/ellipsoids.rst @@ -0,0 +1,159 @@ +.. _ellipsoids: + +================================================================================ +Ellipsoids +================================================================================ +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. + + +.. figure:: ../../images/general_ellipsoid.png + :width: 500 px + :align: center + :alt: Global and local fitting of the ellipsoid + + Global and local fitting of the ellipsoid + + +A complete ellipsoid definition comprises a size (primary) and a shape (secondary) +parameter. + +.. _ellipsoid_size_parameters: + +Ellipsoid size parameters ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +.. option:: +R=<value> + + Radius of the sphere, :math:`R`. + +.. option:: +a=<value> + + Semi-major axis of the ellipsoid, :math:`a`. + +.. _ellipsoid_shape_parameters: + +Ellipsoid shape parameters ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +.. 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. + +.. _ellipsoid_spherification_parameters: + +Ellipsoid spherification parameters ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +.. 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. + +.. _ellipsoid_definitions: + +Built-in ellipsoid definitions ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +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`` + +.. _ellipsoid_transformation_examples: + +Transformation examples ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +Spherical earth with radius 7000km: + +:: + + proj=laton R=7000000 + +Using the GRS80 ellipsoid: + +:: + + proj=laton ellps=GRS80 + +Expressing ellipsoid by semi-major axis and reverse flattening (:math:`1/f`): + +:: + + proj=laton a=6378137.0 rf=298.25 + +Spherical earth based on volume of ellipsoid + +:: + + proj=laton a=6378137.0 rf=298.25 +R_V + |