A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a
quadric surface obtained by
rotating an
ellipse about one of its principal axes; in other words, an
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...
with two equal
semi-diameter
In geometry, the semidiameter or semi-diameter of a set of points may be one half of its diameter; or, sometimes, one half of its extent along a particular direction.
Special cases
The semi-diameter of a sphere, circle, or interval is the sam ...
s. A spheroid has
circular symmetry.
If the ellipse is rotated about its
major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the ...
, 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), also known as gridiron, is a team sport played by two teams of eleven players on a rectangular field with goalposts at each end. The offense, the team wi ...
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
The lentil (''Lens culinaris'' or ''Lens esculenta'') is an edible legume. It is an annual plant known for its lens-shaped seeds. It is about tall, and the seeds grow in pods, usually with two seeds in each. As a food crop, the largest p ...
or a plain
M&M. If the generating ellipse is a circle, the result is a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
.
Due to the combined effects of
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
and
rotation, the
figure of the Earth (and of all
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
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 grc, χάρτης , "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 i ...
and
geodesy the Earth is often approximated by an oblate spheroid, known as the
reference ellipsoid, instead of a sphere. The current
World Geodetic System model uses a spheroid whose radius is at the
Equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can also ...
and at the
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in ...
.
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 qua ...
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
Area
An oblate spheroid with has
surface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
:
The oblate spheroid is generated by rotation about the -axis of an ellipse with semi-major axis and semi-minor axis , therefore may be identified as the
eccentricity. (See
ellipse.)
A prolate spheroid with has surface area
:
The prolate spheroid is generated by rotation about the -axis of an ellipse with semi-major axis and semi-minor axis ; therefore, may again be identified as the
eccentricity. (See
ellipse.)
These formulas are identical in the sense that the formula for can be used to calculate the surface area of a prolate spheroid and vice versa. However, then becomes
imaginary and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.
Volume
The volume inside a spheroid (of any kind) is
:
If is the equatorial diameter, and is the polar diameter, the volume is
:
Curvature
Let a spheroid be parameterized as
:
where is the ''reduced latitude'' or ''
parametric latitude'', is the
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
, and and . Then, the spheroid's
Gaussian curvature is
:
and its
mean curvature is
:
Both of these curvatures are always positive, so that every point on a spheroid is elliptic.
Aspect ratio
The
aspect ratio of an oblate spheroid/ellipse, , is the ratio of the polar to equatorial lengths, while the
flattening
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
(also called oblateness) , is the ratio of the equatorial-polar length difference to the equatorial length:
:
The first
eccentricity (usually simply eccentricity, as above) is often used instead of flattening. It is defined by:
:
The relations between eccentricity and flattening are:
:
All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving the aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an ellipsoidal definition considers its own values to be exact in the form it gives.
Applications
The most common shapes for the density distribution of protons and neutrons in an
atomic nucleus
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...
are
spherical, prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
vector). Deformed nuclear shapes occur as a result of the competition between
electromagnetic repulsion between protons,
surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
and
quantum shell effects.
Oblate spheroids
The oblate spheroid is the approximate shape of rotating
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s and other
celestial bodies, including Earth,
Saturn,
Jupiter
Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousand ...
, and the quickly spinning star
Altair
Altair is the brightest star in the constellation of Aquila and the twelfth-brightest star in the night sky. It has the Bayer designation Alpha Aquilae, which is Latinised from α Aquilae and abbreviated Alpha Aql o ...
. Saturn is the most oblate planet in the
Solar System
The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
, with a
flattening
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
of 0.09796. See
planetary flattening and
equatorial bulge for details.
Enlightenment scientist
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
, working from
Jean Richer
Jean Richer (1630–1696) was a French astronomer and assistant (''élève astronome'') at the French Academy of Sciences, under the direction of Giovanni Domenico Cassini.
Between 1671 and 1673 he performed experiments and carried out celestia ...
's pendulum experiments and
Christiaan Huygens's theories for their interpretation, reasoned that Jupiter and
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
are oblate spheroids owing to their
centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
. Earth's diverse cartographic and geodetic systems are based on
reference ellipsoids, all of which are oblate.
Prolate spheroids
The prolate spheroid is the approximate shape of the ball in several sports, such as in the
rugby ball.
Several
moons of the Solar System approximate prolate spheroids in shape, though they are actually
triaxial ellipsoids. Examples are
Saturn's satellites
Mimas,
Enceladus, and
Tethys and
Uranus' satellite
Miranda.
In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via
tidal forces when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moon
Io, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense
volcanism. The major axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary.
The term is also used to describe the shape of some
nebulae such as the
Crab Nebula.
Fresnel zones, used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver.
The
atomic nuclei
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron ...
of the
actinide
The actinide () or actinoid () series encompasses the 15 metallic chemical elements with atomic numbers from 89 to 103, actinium through lawrencium. The actinide series derives its name from the first element in the series, actinium. The info ...
and
lanthanide elements are shaped like prolate spheroids. In anatomy, near-spheroid organs such as
testis may be measured by their
long and short axes.
Many submarines have a shape which can be described as prolate spheroid.
Dynamical properties
For a spheroid having uniform density, the
moment of inertia is that of an ellipsoid with an additional axis of symmetry. Given a description of a spheroid as having a
major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the ...
, and minor axes , the moments of inertia along these principal axes are , , and . However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:
:
where is the mass of the body defined as
:
See also
*
Ellipsoidal dome
An ellipsoidal dome is a dome (also see geodesic dome), which has a bottom cross-section which is a circle, but has a cupola whose curve is an ellipse.
There are two types of ellipsoidal domes: ''prolate ellipsoidal domes'' and ''oblate ellip ...
*
Equatorial bulge
*
Great ellipse
*
Lentoid
Lentoid is a geometric shape of a three-dimensional body, best described as a circle viewed from one direction and a convex lens viewed from every orthogonal direction. It has no strict mathematical definition, but may be described as the volume e ...
*
Oblate spheroidal coordinates
*
Ovoid
*
Prolate spheroidal coordinates
Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are locat ...
*
Rotation of axes
*
Translation of axes
References
External links
*
* {{Cite EB1911, wstitle=Spheroid, short=1
Surfaces
Quadrics