What
do the terms geoid, ellipsoid, spheroid and datum mean,
and how are they 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.
NOAA
Geoid Index
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 semi-major axis is half the length of the major
axis. The semi-minor axis is half the length of the
minor axis.
For the earth, the semi-major axis is the radius from
the center of the earth to the equator, while the semi-minor
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 semi-major and semi-minor axes.
For example, compare the Clarke 1866 spheroid with the
GRS 1980 spheroid and the WGS 1984 spheroid, based on
the measurements (in meters) below.
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. Since this
doesn't reflect reality very well, a local datum permits
local variations in elevation to be incorporated.
The underlying datum and spheroid to which coordinates
for a dataset are projected 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 X-Coordinate Y-Coordinate
NAD_1927 -122.466903686523 48.7440490722656
NAD_1983 -122.46818353793 48.7438798543649
WGS_1984 -122.46818353793 48.7438798534299
The X-Coordinate 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 Y-Coordinate 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. This is how the coordinates
change based on the datum.
Datum transformations work in either direction. For
example, the transformation NAD_1927_to_NAD_1983_NADCON
will transform from NAD1927 to NAD1983, as well as from
NAD1983 to NAD1927. 
