View source for Ellipsoid

{{for|rotating equilibrium ellipsoid|Jacobi ellipsoid}}
[[File:Ellipsoide.svg|400px|thumb|Examples of ellipsoids with equation <math>{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1:</math><br />
''[[sphere]]'' (top, a=b=c=4),<br />
''[[spheroid]]'' (bottom left, a=b=5, c=3),<br />
''tri-axial'' ellipsoid (bottom right, a=4.5, b=6, c=3)]]

An '''ellipsoid''' is a surface that may be obtained from a [[sphere]] by deforming it by means of directional [[scaling (geometry)|scaling]]s, or more generally, of an [[affine transformation]].

An ellipsoid is a [[quadric surface]];  that is, a [[surface (mathematics)|surface]] that may be defined as the [[zero set]] of a [[polynomial]] of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar [[cross section (geometry)|cross section]] is either an [[ellipse]], or is empty, or is reduced to a single point (this explains the name, meaning "ellipse like"). It is [[bounded set|bounded]], which means that it may be enclosed in a sufficiently large [[sphere]].

An ellipsoid has three pairwise [[perpendicular]] [[rotational symmetry|axes of symmetry]] which intersect at a [[central symmetry|center of symmetry]], called the center of the ellipsoid. The [[line segment]]s that are delimited on the axes of symmetry by the ellipsoid are called the ''principal axes'', or simply axes of the ellipsoid. If the three axes have different lengths, the ellipsoid is said to be ''tri-axial'' or rarely ''scalene'', and the axes are uniquely defined.

If two of the axes have the same length, then the ellipsoid is an ellipsoid of [[surface of revolution|revolution]], also called a [[spheroid]]. In this case, the ellipsoid is invariant under a [[rotation]] around the third axis, and there are thus infinitely many ways of choosing the two perpendicular axes of the same length. If the third axis is shorter, the ellipsoid is an [[oblate spheroid]]; if it is longer, it is a [[prolate spheroid]]. If the three axes have the same length, the ellipsoid is a [[sphere]].

==Standard equation==
Using a [[Cartesian coordinate system]] in which the origin is the center of the ellipsoid and the coordinate axes are axes of the ellipsoid, the [[implicit surface|implicit equation]] of the ellipsoid has the standard form
:<math>{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1,</math>
where {{math|''a'', ''b'', ''c''}} are positive [[real number]]s.

The points {{math|(''a'', 0, 0)}}, {{math|(0, ''b'', 0)}} and {{math|(0, 0, ''c'')}} lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because {{math|''a'', ''b'', ''c''}} are half the length of the principal axes. They correspond to the [[semi-major axis]] and [[semi-minor axis]] of an [[ellipse]].

If <math>a=b>c,</math> one has an [[oblate spheroid]]; if <math>a=b<c,</math> one has a [[prolate spheroid]]; if <math>a=b=c,</math> one has a [[sphere]].

== Parameterization  ==
The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is
:<math>\begin{align}
x&=a\sin(\theta)\cos(\varphi),\\
y&=b\sin(\theta)\sin(\varphi),\\
z&=c\cos(\theta),\end{align}\,\!</math>
where
:<math>
0 \le \theta \le \pi,
\qquad
0 \le \varphi < 2\pi.
</math>
These parameters may be interpreted as [[spherical coordinates]], where <math>\theta</math> is the polar angle and <math>\varphi</math> is the azimuth angle of the point {{math|(''x'', ''y'', ''z'')}} of the ellipsoid.<ref>{{harvtxt|Kreyszig|1972|pp=455–456}}</ref>

==Volume and surface area==
===Volume===
The [[volume]] bounded by the ellipsoid is 
::<math>V=\frac{4}{3}\pi abc.</math>

Alternatively expressed, where A, B and C are the lengths of the principal axes (A=2a, B=2b and C=2c):
::<math>V=\frac\pi6 ABC\approx0.523 ABC</math>.

Note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an [[oblate spheroid|oblate]] or [[prolate spheroid|prolate]] [[spheroid]] when two of them are equal.

The [[volume]] of an ellipsoid is <math>\frac{2}{3}</math> the volume of a [[circumscribed]] [[elliptic cylinder]], and <math>\frac{\pi}{6}</math> the volume of the circumscribed box.

The [[volumes]] of the [[inscribed]] and circumscribed [[rectangular cuboid|boxes]] are respectively:

::<math>V_\text{inscribed} =\frac{8}{3\sqrt3} abc, \qquad V_\text{circumscribed} = 8abc.</math>

===Surface area===
The  [[surface area]] of a general (tri-axial) ellipsoid is<ref>F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors, 2010, NIST  Handbook of Mathematical Functions (Cambridge University Press), available on line at  {{cite web |url=http://dlmf.nist.gov/19.33 |title=Archived copy |accessdate=2012-01-08 |url-status=live |archiveurl=https://web.archive.org/web/20121202103155/http://dlmf.nist.gov/19.33 |archivedate=2012-12-02 }} (see next reference).</ref><ref>NIST (National Institute of Standards and Technology) at http://www.nist.gov {{webarchive|url=https://web.archive.org/web/20150617054320/http://www.nist.gov/ |date=2015-06-17 }}</ref> 
::<math>
S=2\pi c^2+ \frac{2\pi ab}{\sin(\varphi)}
\left(E(\varphi,k)\, \sin^2(\varphi)  + F(\varphi,k)\, \cos^2(\varphi) \right),
</math>
::where
::<math> 
\cos(\varphi) = \frac{c}{a}, \qquad
k^2 =\frac{a^2(b^2-c^2)}{b^2(a^2-c^2)}, \qquad
a\ge b \ge c,
</math>
and where F(φ,k) and E(φ,k) are incomplete [[elliptic integral]]s of the first and second kind respectively.[http://dlmf.nist.gov/19.2]

The surface area of an ellipsoid of revolution (or [[spheroid]]) may be expressed in terms of [[elementary function]]s:
::<math>S_{\text {oblate}} =  2\pi a^2\left(1+\frac{c^2}{e a^2}\cdot \mbox{arctanh}(e)\right)
\quad\mbox{where}\quad e^2=1-\frac{c^2}{a^2}\quad(c<a), </math>
::<math>S_\text{prolate} =  2\pi a^2\left(1+\frac{c}{ae}\cdot \arcsin(e)\right)
\quad\qquad\mbox{where}\;\quad e^2=1-\frac{a^2}{c^2}\quad(c>a), </math>
which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for <math>S_{\rm oblate}</math> can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases ''e'' may again be identified as the [[eccentricity (mathematics)|eccentricity]] of the ellipse formed by the cross section through the symmetry axis. (See [[ellipse]]). Derivations of these results may be found in standard sources, for example [[Mathworld]].<ref>{{cite web|url=http://mathworld.wolfram.com/ProlateSpheroid.html|title=Prolate Spheroid|first=Weisstein, Eric|last=W.|website=mathworld.wolfram.com|accessdate=25 March 2018|url-status=live|archiveurl=https://web.archive.org/web/20170803085757/http://mathworld.wolfram.com/ProlateSpheroid.html|archivedate=3 August 2017}}</ref>

====Approximate formula====
:: <math>S\approx 4\pi \sqrt[p]{\frac{a^p b^p+a^p c^p+b^p c^p}{3}}.\,\!</math>
Here {{nowrap|''p'' ≈ 1.6075}} yields a relative error of at most 1.061%;<ref>[http://www.numericana.com/answer/ellipsoid.htm#thomsen  Final answers] {{webarchive|url=https://web.archive.org/web/20110930084035/http://www.numericana.com/answer/ellipsoid.htm |date=2011-09-30 }}  by Gerard P. Michon (2004-05-13). See Thomsen's formulas and  Cantrell's comments.</ref> a value of {{nowrap|1=''p'' = 8/5 = 1.6}} is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.

In the "flat" limit of ''c'' much smaller than ''a'', ''b'', the area is approximately 2{{pi}}''ab'', equivalent to {{nowrap|1=''p'' ≈ 1.5850}}.

== Plane sections ==
=== Properties ===
[[File:Ellipsoid-ebener-Schnitt.svg|300px|thumb|Plane section of an ellipsoid]]
The intersection of a plane and a sphere is a circle (or is reduced to a single point, or is empty). Any ellipsoid is the image of the unit sphere under some affine transformation, and any plane is the image of some other plane under the same transformation. So, because affine transformations map circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty.<ref>{{citation|first=Abraham Adrian|last=Albert|title=Solid Analytic Geometry|year=2016|origyear=1949|publisher=Dover|isbn=978-0-486-81026-3|page=117}}</ref> Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see [[Circular section]]).

=== Determining the ellipse of a plane section ===
[[File:Ellipso-eb-beisp.svg|thumb|Plane section of an ellipsoid (See example)]]
''Given:'' Ellipsoid <math> \ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\ </math> and the plane with equation <math> \ n_x x+n_y y+n_z z=d \ , </math> which have an ellipse in common.<br>
''Wanted:'' Three vectors <math>\vec f_0</math> (center) and <math>\vec f_1, \; \vec f_2</math> (conjugate vectors), such that the ellipse can be represented by the parametric equation 
:<math>\vec x=\vec f_0 + \vec f_1\cos t+\vec f_2\sin t\quad </math> (See [[Ellipse#Ellipse as an affine image of the unit circle x²+y²=1|ellipse]]).
[[File:Ellipso-eb-ku.svg|300px|thumb|Plane section of the unit sphere (See example)]]
''Solution:'' The scaling <math>\ u=\frac{x}{a}\ ,\ v= \frac{y}{b}\,\ w=\frac{z}{c}\ </math> transforms the ellipsoid onto the unit sphere <math> u^2+v^2+w^2=1 \ </math> and the given plane onto the plane with equation <math> \ n_x au + n_y bv + n_z cw=d \  </math>. Let <math> \ m_u u+m_v v+m_w w=\delta \  </math> be the [[Hesse normal form]] of the new plane and  <math> \; \vec m =(m_u,m_v,m_w)^T \; </math> its unit normal vector. Hence <br> 
<math> \vec e_0= \delta\; \vec m\; </math> is the  ''center'' of the intersection circle and <math>\;\rho=\sqrt{1-\delta^2}\;  </math> its radius (See diagram).<br>
In case of <math>\ m_w=\pm1\ </math> let be <math>\quad \vec e_1=(\rho,0,0)^T\; ,\ \vec e_2=(0,\rho,0)^T\; .</math> (The plane is horizontal !)<br>
In case of <math>\ m_w\neq\pm1\ </math> let be <math>\quad \vec e_1= \rho\, \frac{(m_v,-m_u,0)^T}{\sqrt{m_u^2+m_v^2}}\; ,\ \vec e_2=\vec m\times \vec e_1\ .</math><br>
In any case the vectors  <math> \vec e_1,\vec e_2</math> are orthogonal, parallel to the intersection plane and have length <math> \rho </math> (radius of the circle). Hence the intersection circle can be described by the parametric equation <math>\; \vec u=\vec e_0 + \vec e_1\cos t+\vec e_2\sin t\; .</math><br>
The reverse scaling (See above) transforms the unit sphere back to the ellipsoid and the vectors <math> \vec e_0,\vec e_1,\vec e_2</math> are mapped onto vectors <math>\vec f_0,\vec f_1,\vec f_2</math>, which were wanted  for the parametric representation of the intersection ellipse. <br>
How to find the vertices and semi-axes of the ellipse  is described in [[Ellipse#Ellipse as an affine image of the unit circle x²+y²=1|ellipse]].

''Example:'' The diagrams show an ellipsoid with the semi-axes <math>\; a=4,\; b=5,\; c=3\; </math> which is cut by the plane <math>\; x+y+z=5 \; .</math>

== In general position ==
=== As quadric ===
More generally, an arbitrarily oriented ellipsoid, centered at '''v''', is defined by the solutions '''x''' to the equation
:<math>(\mathbf{x-v})^\mathrm{T}\! A\, (\mathbf{x-v}) = 1,</math>
where ''A'' is a [[positive definite matrix]] and '''x''', '''v''' are [[euclidean vector|vectors]].

The [[eigenvector]]s of ''A'' define the principal axes of the ellipsoid and the [[eigenvalue]]s of ''A'' are the reciprocals of the squares of the semi-axes: <math>a^{-2}</math>, <math>b^{-2}</math> and <math>c^{-2}</math>.<ref>{{cite web |url=http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf |title=Archived copy |accessdate=2013-10-12 |url-status=live |archiveurl=https://web.archive.org/web/20130626233838/http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf |archivedate=2013-06-26 }} pp. 17–18.</ref>
An invertible [[linear transformation]] applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable [[rotation]], a consequence of the [[polar decomposition]] (also, see [[spectral theorem]]). If the linear transformation is represented by a [[symmetric matrix|symmetric 3-by-3 matrix]], then the eigenvectors of the matrix are orthogonal (due to the spectral theorem) and represent the directions of the axes of the ellipsoid; the lengths of the semi-axes are computed from the eigenvalues. The [[singular value decomposition]] and [[polar decomposition]] are matrix decompositions closely related to these geometric observations.

=== Parametric representation ===
[[File:Ellipsoid-affin.svg|300px|thumb|ellipsoid as an affine image of the unit sphere]]
The key to a parametric representation of an ellipsoid in general position is the alternative definition:
*An ellipsoid is an affine image of the unit sphere.
An [[affine transformation]] can be represented by a translation with a vector <math>\vec f_0</math> and a regular 3×3-matrix <math>A</math>:
: <math>\vec x \mapsto \vec f_0 + A\vec x =\vec f_0 + x\vec f_1+y\vec f_2+z\vec f_3</math>,
where <math>\vec f_1, \vec f_2,\vec f_3 </math> are the column vectors of matrix <math>A</math>.

A parametric representation of an ellipsoid in general position can be obtained 
by the parametric representation of a unit sphere (see above) and an affine transformation:
*<math>\vec x(\theta,\varphi)= \vec f_0 + \vec f_1 \cos \theta \cos \varphi+ \vec f_2 \cos \theta \sin \varphi+ \vec f_3 \sin \theta, \quad -\pi/2 \le \theta \le \pi/2, \ 0 \le \varphi < 2 \pi</math>.

If the vectors <math>\vec f_1, \vec f_2, \vec f_3</math> form an orthogonal system, the points with vectors <math>\vec f_0\pm \vec f_i,\ i=1,2,3,</math> are the vertices of the ellipsoid and <math>|\vec f_1|, |\vec f_2|, |\vec f_3|</math> are the semi principal axes.

A surface normal vector at point <math>\vec x(\theta,\varphi)</math> is
: <math>\vec n(\theta,\varphi)=\vec f_2\times\vec f_3\cos \theta \cos \varphi + \vec f_3\times\vec f_1\cos \theta \sin \varphi +\vec f_1\times\vec f_2\sin \theta.</math>

For any ellipsoid there exists an [[Implicit surface|implicit representation]] <math>F(x,y,z)=0</math>. If for simplicity the center of the ellipsoid is the origin, i.e. <math>\vec f_0=(0,0,0)^T</math>, the following equation describes the ellipsoid above:<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf ''Computerunterstützte Darstellende und Konstruktive Geometrie.''] {{webarchive|url=https://web.archive.org/web/20131110190049/http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf |date=2013-11-10 }} Uni Darmstadt (PDF; 3,4&nbsp;MB), S.&nbsp;88.</ref>
: <math>F(x,y,z)= \operatorname{det}(\vec x, \vec f_2, \vec f_3)^2\; +\; \operatorname{det}(\vec f_1,\vec x, \vec f_3)^2\; +\; \operatorname{det}(\vec f_1, \vec f_2,\vec x)^2\; -\; \operatorname{det}(\vec f_1, \vec f_2, \vec f_3)^2 = 0</math>

==Applications==

The ellipsoidal shape finds many practical applications:

;Geodesy
* [[Earth ellipsoid]], a mathematical figure approximating the shape of the Earth.
* [[Reference ellipsoid]], a mathematical figure approximating the shape of planetary bodies in general.

;Mechanics
* [[Poinsot's ellipsoid]], a geometrical method for visualizing the torque-free motion of a rotating [[rigid body]].
* [[Lamé's stress ellipsoid]], an alternative to Mohr's circle for the graphical representation of the stress state at a point.
* [[Manipulability ellipsoid]], used to describe a robot's freedom of motion.

;Crystallography
* [[Index ellipsoid]], a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal.
* [[Thermal ellipsoid]], ellipsoids used in crystallography to indicate the magnitudes and directions of the thermal vibration of atoms in crystal structures.

;Lighting
* [[Ellipsoidal reflector floodlight]]
* [[Ellipsoidal reflector spotlight]]

;Medicine
* Measurements obtained from [[MRI]] imaging of the [[prostate]] can be used to determine the volume of the gland using the approximation L × W × H × 0.52 (where 0.52 is an approximation for π/6) <ref>{{cite journal|last1=Bezinque|first1=Adam|display-authors=etal|title=Determination of Prostate Volume: A Comparison of Contemporary Methods|journal=Academic Radiology|volume=25|issue=12|pages=1582–1587|doi=10.1016/j.acra.2018.03.014|pmid=29609953|year=2018}}</ref>

===Dynamical properties===

The [[mass]] of an ellipsoid of uniform density ρ is:
:<math>m = \rho V = \rho \frac{4}{3} \pi abc.\,\!</math>

The  [[Moment of Inertia|moments of inertia]] of an ellipsoid of uniform density are:
::<math>I_{\mathrm{xx}} = \frac{m(b^2+c^2)}{5},\qquad
I_{\mathrm{yy}} = \frac{m(c^2+a^2)}{5},\qquad
I_{\mathrm{zz}} = \frac{m(a^2+b^2)}{5},</math>

::<math> I_{\mathrm{xy}}=  I_{\mathrm{yz}} = I_{\mathrm{zx}} =0.\,\!</math>

For <math>a = b = c</math> these moments of inertia reduce to those for a sphere of uniform density.

[[File:2003EL61art.jpg|right|thumb|Artist's conception of {{dp|Haumea}}, a Jacobi-ellipsoid [[dwarf planet]], with its two moons]]
Ellipsoids and [[cuboid]]s rotate stably along their major or minor axes, but not along their median axis. This can be seen experimentally by throwing an eraser with some spin. In addition, [[moment of inertia]] considerations mean that rotation along the major axis is more easily perturbed than rotation along the minor axis.<ref>Goldstein, H G (1980). ''Classical Mechanics'', (2nd edition) Chapter 5.</ref>

One practical effect of this is that scalene astronomical bodies such as {{dp|Haumea}} generally rotate along their minor axes (as does Earth, which is merely oblate); in addition, because of [[tidal locking]], moons in [[synchronous orbit]] such as [[Mimas (moon)|Mimas]] orbit with their major axis aligned radially to their planet.

A spinning body of homogeneous self-gravitating fluid will assume the form of either a [[Maclaurin spheroid]] (oblate spheroid) or [[Jacobi ellipsoid]] (scalene ellipsoid) when in [[hydrostatic equilibrium]], and for moderate rates of rotation.  At faster rotations, non-ellipsoidal [[:wikt:pyriform|piriform]] or [[oviform]] shapes can be expected, but these are not stable.

===Fluid dynamics===

The ellipsoid is the most general shape for which it has been possible to calculate the [[creeping flow]] of fluid around the solid shape. The calculations include the force required to translate through a fluid and to rotate within it. Applications include determining the size and shape of large molecules, the sinking rate of small particles, and the swimming abilities of [[microorganisms]].<ref>Dusenbery, David B. (2009).''Living at Micro Scale'', Harvard University Press, Cambridge, Massachusetts {{isbn|978-0-674-03116-6}}.</ref>

===In probability and statistics===

The [[elliptical distribution]]s, which generalize the [[multivariate normal distribution]] and are used in [[finance]], can be defined in terms of their [[density function]]s.  When they exist, the density functions ''f'' have the structure:
:<math>f(x)= k \cdot g((x-\mu)\Sigma^{-1}(x-\mu)^\mathrm{T})</math>
where <math>k</math> is a scale factor, <math>x</math> is an <math>n</math>-dimensional [[random vector|random row vector]] with median vector <math>\mu</math> (which is also the mean vector if the latter exists),  <math>\Sigma</math> is a [[positive definite matrix]] which is proportional to the [[covariance matrix]] if the latter exists, and <math>g</math> is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.<ref>Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.</ref> The multivariate normal distribution is the special case in which <math>g(z)=e^{-z/2}</math> for quadratic form <math>z</math>.

Thus the density function is a scalar-to-scalar transformation of a quadric expression. Moreover, the equation for any [[contour line|iso-density surface]] states that the quadric expression equals some constant specific to that value of the density, and the iso-density surface is an ellipsoid.

== In higher dimensions ==
The volume of a higher-dimensional ellipsoid (a ''hyperellipsoid'') can be calculated using the dimensional constant given for the [[Volume of an n-ball|volume of a hypersphere]].
One can also define hyperellipsoids as the images of spheres under invertible linear transformations. The spectral theorem can again be used to obtain a standard equation akin to the one given above.

== See also ==
* [[Ellipsoid method]]
* [[Ellipsoidal coordinates]]
* [[Elliptical distribution]], in statistics
* [[Flattening]], also called ''[[ellipticity]]'' and ''[[oblateness]]'', 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.
* [[Focaloid]], a shell bounded by two concentric, confocal ellipsoids
* [[Geodetic datum]], the gravitational Earth modeled by a best fitted elipsoid
* [[Homoeoid]], a shell bounded by two concentric, similar ellipsoids
* [[List of surfaces]]

== Notes ==
<references />

== References ==
* {{citation | last1 = Kreyszig | first1 = Erwin | authorlink = Erwin Kreyszig | title = Advanced Engineering Mathematics | edition = 3rd | location = New York | publisher = [[John Wiley & Sons|Wiley]] | year = 1972 | isbn = 0-471-50728-8 | url-access = registration | url = https://archive.org/details/advancedengineer00krey }}

== External links ==
{{commons category|Ellipsoids}}
* "[http://demonstrations.wolfram.com/Ellipsoid/ Ellipsoid]" by Jeff Bryant, [[Wolfram Demonstrations Project]], 2007.
* [http://mathworld.wolfram.com/Ellipsoid.html Ellipsoid] and [http://mathworld.wolfram.com/QuadraticSurface.html Quadratic Surface], [[MathWorld]].

[[Category:Geometric shapes]]
[[Category:Surfaces]]
[[Category:Quadrics]]

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