# The geoid, ellipsoid, spheroid and datum, and how they are related

The geoid is defined as the surface of the earth's gravity field, which is approximately the same as mean sea level. It is perpendicular to the direction of gravity pull. Since the mass of the earth is not uniform at all points, and the direction of gravity changes, the shape of the geoid is irregular.

Click on the link below to access a website maintained by the National Oceanographic & Atmospheric Administration (NOAA). The website has links to images showing interpretations of the geoid under North America. http://www.ngs.noaa.gov/GEOID/GEOID96/geo-indx.html

To simplify the model, various spheroids or ellipsoids have been devised. These terms are used interchangeably. For the remainder of this article, the term spheroid will be used.

A spheroid is a three-dimensional shape created from a two-dimensional ellipse. The ellipse is an oval, with a major axis (the longer axis), and a minor axis (the shorter axis). If you rotate the ellipse, the shape of the rotated figure is the spheroid.

The semimajor axis is half the length of the major axis. The semiminor axis is half the length of the minor axis.

For the earth, the semimajor axis is the radius from the center of the earth to the equator, while the semiminor axis is the radius from the center of the earth to the pole.

One particular spheroid is distinguished from another by the lengths of the semimajor and semiminor axes. For example, compare the Clarke 1866 spheroid with the GRS 1980 and the WGS 1984 spheroids, based on the measurements (in meters) below.

Spheroid |
Semimajor axis (m) |
Semiminor axis (m) |
---|---|---|

Clarke 1866 |
6378206.4 |
6356583.8 |

GRS80 1980 |
6378137 |
6356752.31414 |

WGS84 1984 |
6378137 |
6356752.31424518 |

A particular spheroid can be selected for use in a specific geographic area, because that particular spheroid does an exceptionally good job of mimicking the geoid for that part of the world. For North America, the spheroid of choice is GRS 1980, on which the North American Datum 1983 (NAD83) is based.

A datum is built on top of the selected spheroid, and can incorporate local variations in elevation. With the spheroid, the rotation of the ellipse creates a totally smooth surface across the world. Because this doesn't reflect reality very well, a local datum can incorporate local variations in elevation.

The underlying datum and spheroid to which coordinates for a dataset are referenced can change the coordinate values. An illustrative example using the city of Bellingham, Washington follows. Compare the coordinates in decimal degrees for Bellingham using NAD27, NAD83 and WGS84. It is apparent that while NAD83 and WGS84 express coordinates that are nearly identical, NAD27 is quite different, because the underlying shape of the earth is expressed differently by the datums and spheroids used.

Datum |
Longitude |
Latitude |
---|---|---|

NAD 1927 |
-122.46690368652 |
48.7440490722656 |

NAD 1983 |
-122.46818353793 |
48.7438798543649 |

WGS 1984 |
-122.46818353793 |
48.7438798534299 |

The longitude is the measurement of the angle from the prime meridian at Greenwich, England, to the center of the earth, then west to the longitude of Bellingham, Washington. The latitude is the measurement of the angle formed from the equator to the center of the earth, then north to the latitude of Bellingham, Washington.

If the surface of the earth at Bellingham is bulged out, the angular measurements in decimal degrees from Greenwich and the equator will become slightly larger. If the surface at Bellingham is lowered, the angles will become slightly smaller. These are two examples of how the coordinates change based on the datum.