Dictionary Definition
ellipsoid adj : in the form of an ellipse [syn:
ellipsoidal,
spheroidal, noncircular]
n : a surface whose plane sections are all ellipses or circles;
"the Earth is an ellipsoid"
User Contributed Dictionary
English
Noun
 a surface, all of whose cross sections are elliptic or circular (includes the sphere)
Derived terms
Translations
 Finnish: ellipsoidi
 Swedish: ellipsoid
Adjective
 of or pertaining to an ellipse; ellipsoidal
 Shaped like an ellipse; elliptical.
 Shaped like a symmetrical oval that is evenly tapered on both ends.
Derived terms
Swedish
Noun
ellipsoidExtensive Definition
An ellipsoid is a type of quadric surface
that is a higher dimensional
analogue of an ellipse.
The equation of a standard ellipsoid body in an xyzCartesian
coordinate system is
 ++=1
where a and b are the equatorial radii (along the
x and y axes) and c is the polar radius (along the zaxis), all of
which are fixed
positive real numbers
determining the shape of the ellipsoid.
If all three radii are equal, the solid body is a
sphere; if two radii are
equal, the ellipsoid is a spheroid:
The points (a,0,0), (0,b,0) and (0,0,c) lie on
the surface and the line segments from the origin to these points
are called the semiprincipal axes. These correspond to the
semimajor
axis and semiminor
axis of the appropriate ellipses.
Parameterization
Using the common coordinates, where \beta\,\! is a point's reduced, or parametric latitude and \!\!\!\lambda\,\! is its planetographic longitude, an ellipsoid can be parameterized by:


 \begin



 \begin90^\circ\leq\beta\leq+90^\circ;


 (Note that this parameterization is not
 11 at the poles, where \scriptstyle\,\!)

Or, using
spherical coordinates, where \!\!\!\theta\,\! is the
colatitude, or zenith, and \!\!\!\varphi\,\! is the longitude in
360°, or azimuth:



 \begin



 \begin0\leq\theta\leq^\circ;
Volume
The volume of an ellipsoid is given by the formula \frac\pi abc.\,\!
Surface area
The surface area of an ellipsoid is given by: 2\pi\left(c^2+b\sqrtE(o\!\varepsilon,m)+\fracF(o\!\varepsilon,m)\right),\,\!
 o\!\varepsilon=\arccos\left(\frac\right)\;\textrm\arccos\left(\frac\right)\;(\textrm),\,\!
Unlike the area of a sphere, the surface area of
a general ellipsoid cannot be expressed exactly by an elementary
function.
An approximate formula is:

 \approx 4\pi\!\left(\frac\right)^.\,\!
Where p ≈ 1.6075 yields a relative error of at
most 1.061% (Knud
Thomsen's formula); a value of p = 8/5 = 1.6 is optimal for
nearly spherical ellipsoids, with a relative error of at most
1.178% (David W.
Cantrell's formula).
Exact formulae can be obtained for the case a = b
(i.e., a spherical equator):
 If oblate: 2\pi\!\left(a^2+c^2\frac\right);\,\!
 If prolate: 2\pi\!\left(a^2+c^2\frac\right);\,\!
In the "flat" limit of c \ll a, b\,\!, the area
is approximately 2\pi ab.\,\!
Mass properties
The mass of an ellipsoid of uniform density is: m = \rho V = \rho \frac \pi abc\,\!
The mass moments
of inertia of an ellipsoid of uniform density are:
 I_ = m
 I_ = m
 I_ = m
where I_\,\!, I_\,\!, and I_\,\! are the moments
of inertia about the x, y, and z axes, respectively. Products
of inertia are zero.
It can easily be shown that if a=b=c, then the
moments of inertia reduce to those for a uniformdensity
sphere.
Conversely, if the mass and principle inertias of
an arbitrary rigid body are known, an equivalent ellipsoid of
uniform density can be constructed, with the following
characteristics:
 a = \sqrt
 b = \sqrt
 c = \sqrt
 \rho = \frac \!
Linear transformations
If one applies an invertible linear transformation to a sphere, one obtains an ellipsoid; it can be brought into the above standard form by a suitable rotation, a consequence of the spectral theorem. If the linear transformation is represented by a symmetric 3by3 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 semiaxes are given by the eigenvalues.One can also define ellipsoids in higher
dimensions, 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.
Egg shape
The shape of a chicken egg is approximately that of half each a prolate and roughly spherical (potentially even minorly oblate) ellipsoid joined at the equator, sharing a principal axis of rotational symmetry. Although the term eggshaped usually implies a lack of reflection symmetry across the equatorial plane, it may also refer to true prolate ellipsoids. It can also be used to describe the 2D figure that, revolved around its major axis, produces the 3D surface. See also oval (geometry).References
 "Ellipsoid" by Jeff Bryant, The Wolfram Demonstrations Project, 2007.
 Ellipsoid and Quadratic Surface, MathWorld.
See also
 Paraboloid
 Hyperboloid
 Reference ellipsoid
 Geoid
 Ellipsoid method
 Superellipsoid
 , an ellipsoid shaped planetoid
External links
ellipsoid in Arabic: سطح ناقص
ellipsoid in Asturian: Elipsoide de
revolución
ellipsoid in Catalan: El·lipsoide
ellipsoid in Czech: Elipsoid
ellipsoid in German: Ellipsoid
ellipsoid in Spanish: Elipsoide
ellipsoid in French: Ellipsoïde
ellipsoid in Hebrew: אליפסואיד
ellipsoid in Italian: Ellissoide
ellipsoid in Japanese: 楕円体
ellipsoid in Dutch: Ellipsoïde
ellipsoid in Norwegian: Ellipsoide
ellipsoid in Norwegian Nynorsk: Ellipsoide
ellipsoid in Polish: Elipsoida
ellipsoid in Portuguese: Elipsóide
ellipsoid in Russian: Эллипсоид
ellipsoid in Simple English: Ellipsoid
ellipsoid in Finnish: Ellipsoidi
ellipsoid in Swedish: Ellipsoid
ellipsoid in Tamil: நீளுருண்டை
ellipsoid in Thai: ทรงรี
ellipsoid in Turkish: Elipsoit
ellipsoid in Vietnamese: Ellipsoid
ellipsoid in Chinese: 椭球