# 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.

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=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
```