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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 :\frac+\frac+\frac = 1. The equation of a spheroid with as the symmetry axis is given by setting : :\frac+\frac=1. 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: :C_\text = 2\pi a 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: C_\text \,=\, 4a\int_0^\sqrt \ d\theta 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: :C_\text = 2\sqrt /math>


Area

An oblate spheroid with has
surface area The surface area (symbol ''A'') 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 d ...
:S_\text = 2\pi a^2\left(1+\frac\operatornamee\right)=2\pi a^2+\pi \frac\ln \left( \frac\right) \qquad \mbox \quad e^2=1-\frac. 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 Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
. (See
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 ...
.) A prolate spheroid with has surface area :S_\text = 2\pi a^2\left(1+\frac\arcsin \, e\right) \qquad \mbox \quad e^2=1-\frac. 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 Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
. (See
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 ...
.) 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 :\tfrac\pi a^2c\approx4.19a^2c. If is the equatorial diameter, and is the polar diameter, the volume is :\tfracA^2C\approx0.523A^2C.


Curvature

Let a spheroid be parameterized as : \boldsymbol\sigma (\beta,\lambda) = (a \cos \beta \cos \lambda, a \cos \beta \sin \lambda, c \sin \beta), 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 lett ...
, and and . Then, the spheroid's
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
is : K(\beta,\lambda) = \frac, and its mean curvature is : H(\beta,\lambda) = \frac. Both of these curvatures are always positive, so that every point on a spheroid is elliptic.


Aspect ratio

The ''
aspect ratio The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...
'' of an oblate spheroid/ellipse, , is the ratio of the polar to equatorial lengths, while the '' flattening'' (also called ''oblateness'') , is the ratio of the equatorial-polar length difference to the equatorial length: :f = \frac = 1 - \frac . The first ''eccentricity'' (usually simply eccentricity, as above) is often used instead of flattening. It is defined by: : e = \sqrt The relations between eccentricity and flattening are: : \begin e &= \sqrt \\ f &= 1 - \sqrt \end 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.


Occurrence and 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 at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
are spherical, prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
vector). Deformed nuclear shapes occur as a result of the competition between
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
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 (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
and quantum shell effects. Spheroids are common in 3D cell cultures. Rotating equilibrium spheroids include the Maclaurin spheroid and the Jacobi ellipsoid.
Spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...
is also a shape of archaeological artifacts.


Oblate spheroids

The oblate spheroid is the approximate shape of rotating
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 and other celestial bodies, including Earth,
Saturn Saturn is the sixth planet from the Sun and the second largest in the Solar System, after Jupiter. It is a gas giant, with an average radius of about 9 times that of Earth. It has an eighth the average density of Earth, but is over 95 tim ...
,
Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...
, and the quickly spinning star Altair. Saturn is the most oblate planet in the
Solar System The Solar SystemCapitalization 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 "Sola ...
, with a flattening of 0.09796. See planetary flattening and equatorial bulge for details. Enlightenment scientist
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, working from Jean Richer's pendulum experiments and
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
'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 Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
are oblate spheroids owing to their centrifugal force. 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 used in
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 ...
and in rugby. Several moons of the Solar System approximate prolate spheroids in shape, though they are actually triaxial ellipsoids. Examples are
Saturn Saturn is the sixth planet from the Sun and the second largest in the Solar System, after Jupiter. It is a gas giant, with an average radius of about 9 times that of Earth. It has an eighth the average density of Earth, but is over 95 tim ...
's satellites Mimas, Enceladus, and Tethys and
Uranus Uranus is the seventh planet from the Sun. It is a gaseous cyan-coloured ice giant. Most of the planet is made of water, ammonia, and methane in a Supercritical fluid, supercritical phase of matter, which astronomy calls "ice" or Volatile ( ...
's satellite Miranda. In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via
tidal forces The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the d ...
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. This combines with the smaller oblate distortion from the synchronous rotation to cause the body to become triaxial. The term is also used to describe the shape of some
nebula A nebula (; or nebulas) is a distinct luminescent part of interstellar medium, which can consist of ionized, neutral, or molecular hydrogen and also cosmic dust. Nebulae are often star-forming regions, such as in the Pillars of Creation in ...
e such as the
Crab Nebula The Crab Nebula (catalogue designations M1, NGC 1952, Taurus A) is a supernova remnant and pulsar wind nebula in the constellation of Taurus (constellation), Taurus. The common name comes from a drawing that somewhat resembled a crab with arm ...
. 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 of the actinide 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 , 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: :\begin A = B &= \tfrac15 M\left(a^2+c^2\right), \\ C &= \tfrac15 M\left(a^2+b^2\right) =\tfrac25 M\left(a^2\right), \end where is the mass of the body defined as : M = \tfrac43 \pi a^2 c\rho.


See also

* Ellipsoidal dome * Equatorial bulge * Great ellipse * Lentoid * Oblate spheroidal coordinates * Ovoid * Prolate spheroidal coordinates * Rotation of axes * Translation of axes


References


External links

* * {{Cite EB1911, wstitle=Spheroid, short=1 Ellipsoids Surfaces of revolution