+
+ +
+

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.

+
+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

+
+
++R=<value>
+

Radius of the sphere, \(R\).

+
+ +
+
++a=<value>
+

Semi-major axis of the ellipsoid, \(a\).

+
+ +
+
+

Ellipsoid shape parameters

+
+
++rf=<value>
+

Reverse flattening of the ellipsoid, \(1/f\).

+
+ +
+
++f=<value>
+

Flattening of the ellipsoid, \(f\).

+
+ +
+
++es=<value>
+

Eccentricity squared, \(e^2\).

+
+ +
+
++e=<value>
+

Eccentricity, \(e\).

+
+ +
+
++b=<value>
+

Semi-minor axis, \(b\).

+
+ +

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

+
+
++R_A=<value>
+

A sphere with the same surface area as the ellipsoid.

+
+ +
+
++R_V=<value>
+

A sphere with the same volume as the ellipsoid.

+
+ +
+
++R_a=<value>
+

A sphere with \(R = (a + b)/2\) (arithmetic mean).

+
+ +
+
++R_g=<value>
+

A sphere with \(R = \sqrt{ab}\) (geometric mean).

+
+ +
+
++R_h=<value>
+

A sphere with \(R = 2ab/(a+b)\) (harmonic mean).

+
+ +
+
++R_lat_a=<phi>
+

A sphere with \(R\) being the arithmetic mean of the corresponding +ellipsoid at latitude \(\phi\).

+
+ +
+
++R_lat_g=<phi>
+

A sphere with \(R\) being the geometric mean of the corresponding +ellipsoid at latitude \(\phi\).

+
+ +

If +R is given as size parameter, any shape and spherification +parameters given are ignored.

+
+
+

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

+
+
+

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 (\(1/f\)):

+
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
+
+
+
+
+ + +
+