The previous discussion of direct geometric map projections assumes that the Earth is a sphere, and for many maps this is satisfactory. However, due to rotation of the Earth around its axis, the planet bulges slightly at the equator. This flattening of the sphere makes it an oblate spheroid, which is an ellipse rotated around its shorter axis.

Ellipse Diagram

An ellipse is defined by its semi-major (long) and semi-minor (short) axes.

The amount of flattening of the Earth is expressed as the ratio:

Where:

a = equatorial radius (semi-major axis)

b = polar radius (semi-minor axis)

Most map projections use eccentricity rather than flattening. The relationship is:

Where:

e2 = eccentricity

The flattening of the Earth is about 1 part in 300, and becomes significant in map accuracy at a scale of 1:100,000 or larger.

Calculation of a map projection requires definition of the spheroid (or ellipsoid) in terms of the length of axes and eccentricity squared (or radius of the reference sphere). Several principal spheroids are in use by one or more countries. Differences are due primarily to calculation of the spheroid for a particular region of the Earthâ€™s surface. Only recently have satellite tracking data provided spheroid determinations for the entire Earth. However, these spheroids may not give the best fit for a particular region. In North America, the spheroid in use is the Clarke 1866 for NAD27 and GRS 1980 for NAD83 (State Plane).

If other regions are to be mapped, different projections should be used. Upon choosing a desired projection type, you have the option to choose a spheroid. The Projection Chooser contains all the supported spheroids. The following lists some of the commonly used spheroids.

- Airy
- Australian National
- Bessel
- Clarke 1866
- Clarke 1880
- Everest
- GRS 1980
- Helmert
- Hough
- International 1909
- Krasovsky
- Mercury 1960
- Modified Airy
- Modified Everest
- Modified Mercury 1968
- New International 1967
- Southeast Asia
- Sphere of Nominal Radius of Earth
- Sphere of Radius 6370977m
- Walbeck
- WGS 66
- WGS 72
- WGS 84

The spheroids listed above are the most commonly used. There are many other spheroids available, and they are listed in Projection Chooser. These additional spheroids are not documented in this Guide. You can add your own map projections and spheroids to ERDAS IMAGINE.

The semi-major and semi-minor axes of supported spheroids are listed in tables Earth Spheroids for use in ERDAS IMAGINE and Non-Earth Spheroids for use in ERDAS IMAGINE, as well as the principal uses of these spheroids.

Earth Spheroids for use in ERDAS IMAGINE

Spheroid | Semi-Major Axis | Semi-Minor Axis | Use |

165 | 6378165.0 | 6356783.0 | Global |

Airy (1940) | 6377563.0 | 6356256.91 | England |

Airy Modified (1849) | Ireland | ||

Australian National (1965) | 6378160.0 | 6356774.719 | Australia |

Bessel (1841) | 6377397.155 | 6356078.96284 | Central Europe, Chile, and Indonesia |

Bessell (Namibia) | 6377483.865 | 6356165.383 | Namibia |

Clarke 1858 | 6378293.0 | 6356619.0 | Global |

Clarke 1866 | 6378206.4 | 6356583.8 | North America and the Philippines |

Clarke 1880 | 6378249.145 | 6356514.86955 | France and Africa |

Clarke 1880 IGN | 6378249.2 | 6356515.0 | Global |

Everest (1830) | 6377276.3452 | 6356075.4133 | India, Burma, and Pakistan |

Everest (1956) | 6377301.243 | 6356100.2284 | India, Nepal |

Everest (1969) | 6377295.664 | 6356094.6679 | Global |

Everest (Malaysia & Singapore) | 6377304.063 | 6356103.038993 | Global |

Everest (Pakistan) | 6377309.613 | 6356108.570542 | Pakistan |

Everest (Sabah & Sarawak) | 6377298.556 | 6356097.5503 | Brunei, East Malaysia |

Fischer (1960) | 6378166.0 | 6356784.2836 | Global |

Fischer (1968) | 6378150.0 | 6356768.3372 | Global |

GRS 1980 (Geodetic Reference System) | 6378137.0 | 6356752.31414 | Adopted in North America for 1983 Earth-centered coordinate system (satellite) |

Hayford | 6378388.0 | 6356911.946128 | Global |

Helmert | 6378200.0 | 6356818.16962789092 | Egypt |

Hough | 6378270.0 | 6356794.343479 | As International 1909 above, with modification of ellipse axes |

IAU 1965 | 6378160.0 | 6356775.0 | Global |

Indonesian 1974 | 6378160.0 | 6356774.504086 | Global |

International 1909 (= Hayford) | 6378388.0 | 6356911.94613 | Remaining parts of the world not listed here |

IUGG 1967 | 6378160.0 | 6356774.516 | Hungary |

Krasovsky (1940) | 6378245.0 | 6356863.0188 | Former Soviet Union and some East European countries |

Mercury 1960 | 6378166.0 | 6356794.283666 | Early satellite, rarely used |

Modified Airy | 6377341.89 | 6356036.143 | As Airy above, more recent version |

Modified Everest | 6377304.063 | 6356103.039 | As Everest above, more recent version |

Modified Mercury 1968 | 6378150.0 | 6356768.337303 | As Mercury 1960 above, more recent calculation |

Modified Fischer (1960) | 6378155.0 | 6356773.3205 | Singapore |

New International 1967 | 6378157.5 | 6356772.2 | As International 1909 below, more recent calculation |

SGS 85 (Soviet Geodetic System 1985) | 6378136.0 | 6356751.3016 | Soviet Union |

South American (1969) | 6378160.0 | 6356774.7192 | South America |

Southeast Asia | 6378155.0 | 6356773.3205 | As named |

Sphere | 6371000.0 | 6371000.0 | Global |

Sphere of Nominal Radius of Earth | 6370997.0 | 6370997.0 | A perfect sphere |

Sphere of Radius 6370997 m | 6370997.0 | 6370997.0 | A perfect sphere with the same surface area as the Clarke 1866 spheroid |

Walbeck (1819) | 6376896.0 | 6355834.8467 | Soviet Union, up to 1910 |

WGS 60 (World Geodetic System 1960) | 6378165.0 | 6356783.287 | Global |

WGS 66 (World Geodetic System 1966) | 6378145.0 | 6356759.769356 | As WGS 72 above, older version |

WGS 72 (World Geodetic System 1972) | 6378135.0 | 6356750.519915 | NASA (satellite) |

WGS 84 (World Geodetic System 1984) | 6378137.0 | 6356752.31424517929 | As WGS 72, more recent calculation |

Non-Earth Spheroids

Spheroid models can be applied to planetary bodies other than the Earth, such as the Moon, Venus, Mars, various asteroids, and other planets in our Solar System. Spheroids for these planetary bodies have a defined semi-major axis and a semi-minor axis, measured in meters, corresponding to Earth spheroids.

See the "Ellipse Diagram" at the top of this topic for an illustration of the axes defined in an ellipse.

The semi-major and semi-minor axes of the supported extraterrestrial spheroids are listed in the following table.

Non-Earth Spheroids for use in ERDAS IMAGINE

Spheroid | Semi-Major Axis | Semi-Minor Axis |

Moon | 1738100.0 | 1736000.0 |

Mercury | 2439700.0 | 2439700.0 |

Venus | 6051800.0 | 6051800.0 |

Mars | 3396200.0 | 3376200.0 |

Jupiter | 71492000.0 | 66854000.0 |

Saturn | 60268000.0 | 54364000.0 |

Uranus | 25559000.0 | 24973000.0 |

Neptune | 24764000.0 | 24341000.0 |

Pluto | 1195000.0 | 1195000.0 |