A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a
quadric surface obtained by
rotating an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
about one of its principal axes; in other words, an
ellipsoid with two equal
semi-diameters. A spheroid has
circular symmetry.
If the ellipse is rotated about its
major axis, the result is a ''prolate spheroid'', elongated like a
rugby ball. The
American football
American football, referred to simply as football in the United States and Canada and also known as gridiron football, is a team sport played by two teams of eleven players on a rectangular American football field, field with goalposts at e ...
is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its
minor axis, the result is an ''oblate spheroid'', flattened like a
lentil or a plain
M&M. If the generating ellipse is a circle, the result is a
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
.
Due to the combined effects of
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
and
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
, the
figure of the Earth
In geodesy, the figure of the Earth is the size and shape used to model planet Earth. The kind of figure depends on application, including the precision needed for the model. A spherical Earth is a well-known historical approximation that is ...
(and of all
planet
A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
s) is not quite a sphere, but instead is slightly
flattened in the direction of its axis of rotation. For that reason, in
cartography
Cartography (; from , 'papyrus, sheet of paper, map'; and , 'write') is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an imagined reality) can ...
and
geodesy
Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...
the Earth is often approximated by an oblate spheroid, known as the
reference ellipsoid, instead of a sphere. The current
World Geodetic System
The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also describ ...
model uses a spheroid whose radius is at the
Equator
The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...
and at the
poles
Pole or poles may refer to:
People
*Poles (people), another term for Polish people, from the country of Poland
* Pole (surname), including a list of people with the name
* Pole (musician) (Stefan Betke, born 1967), German electronic music artist
...
.
The word ''spheroid'' originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape; that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the
Earth's gravity
The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).
It is a vector qu ...
geopotential model).
Equation
The equation of a tri-axial ellipsoid centred at the origin with semi-axes , and aligned along the coordinate axes is
:
The equation of a spheroid with as the
symmetry axis is given by setting :
:
The semi-axis is the equatorial radius of the spheroid, and is the distance from centre to pole along the symmetry axis. There are two possible cases:
* : oblate spheroid
* : prolate spheroid
The case of reduces to a sphere.
Properties
Circumference
The equatorial circumference of a spheroid is measured around its
equator
The equator is the circle of latitude that divides Earth into the Northern Hemisphere, Northern and Southern Hemisphere, Southern Hemispheres of Earth, hemispheres. It is an imaginary line located at 0 degrees latitude, about in circumferen ...
and is given as:
:
The meridional or polar circumference of a spheroid is measured through its
poles
Pole or poles may refer to:
People
*Poles (people), another term for Polish people, from the country of Poland
* Pole (surname), including a list of people with the name
* Pole (musician) (Stefan Betke, born 1967), German electronic music artist
...
and is given as:
The volumetric circumference of a spheroid is the circumference of a
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
of equal volume as the spheroid and is given as:
: