Editing Ellipsoid (section)

==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 body|planetary bodies]] in general.

;[[Mechanics]]
* [[Poinsot's ellipsoid]], a geometrical method for visualizing the [[Torque-free precession|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 (mechanics)|stress]] state at a point.
* [[Manipulability ellipsoid]], used to describe a robot's freedom of motion.
* [[Jacobi ellipsoid]], a triaxial ellipsoid formed by a rotating fluid

;[[Crystallography]]
* [[Index ellipsoid]], a diagram of an ellipsoid that depicts the orientation and relative magnitude of [[Refractive index|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 structure]]s.

=== Computer science ===

* [[Ellipsoid method]], a [[convex optimization]] algorithm of theoretical significance
;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 {{math|''L'' × ''W'' × ''H'' × 0.52}} (where 0.52 is an approximation for {{sfrac|{{math|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|s2cid=4621745 }}</ref>

===Dynamical properties===
The [[mass]] of an ellipsoid of uniform [[density]] {{mvar|ρ}} is
:<math>m = V \rho = \tfrac{4}{3} \pi abc \rho.</math>

The [[Moment of Inertia|moments of inertia]] of an ellipsoid of uniform density are
:<math>\begin{align}
  I_\mathrm{xx} &= \tfrac{1}{5}m\left(b^2 + c^2\right), &
    I_\mathrm{yy} &= \tfrac{1}{5}m\left(c^2 + a^2\right), &
    I_\mathrm{zz} &= \tfrac{1}{5}m\left(a^2 + b^2\right), \\[3pt]
  I_\mathrm{xy} &=  I_\mathrm{yz} = I_\mathrm{zx} = 0.
\end{align}</math>

For {{math|1=''a'' = ''b'' = ''c''}} 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 spheroid|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 {{mvar|f}} have the structure:
:<math>f(x) = k \cdot g\left((\mathbf x - \boldsymbol\mu)\boldsymbol\Sigma^{-1}(\mathbf x - \boldsymbol\mu)^\mathsf{T}\right)</math>

where {{mvar|k}} is a scale factor, {{math|'''x'''}} is an {{mvar|n}}-dimensional [[random vector|random row vector]] with median vector {{math|'''&mu;'''}} (which is also the mean vector if the latter exists), {{math|'''&Sigma;'''}} is a [[positive definite matrix]] which is proportional to the [[covariance matrix]] if the latter exists, and {{mvar|g}} 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'') {{=}} exp(−{{sfrac|''z''|2}})}} for quadratic form {{mvar|z}}.

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.