Spheroids

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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