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| {{About|the geometric three-dimensional shape}}
| | Some consumers of computer or computer are aware which their computer become slower or have several mistakes following using for a while. But most persons don't understand how to accelerate their computer plus a few of them don't dare to operate it. They always find several experts to keep the computer in good condition yet they have to spend several income on it. Actually, you are able to do it by oneself. There are many registry cleaner software which there are 1 of them online. Some of them are free and you just should download them. After installing it, this registry cleaner software will scan the registry. If it found these errors, it usually report we and you can delete them to keep the registry clean. It is simple to work and it is the most effective way to repair registry.<br><br>We want to recognize certain simple and cheap techniques which can resolve the issue of your computer and speed it up. The earlier you fix it, the less damage the computer gets. I can tell about certain helpful techniques which will help you to accelerate we computer.<br><br>The Windows registry is a system database of info. Windows and alternative software shop a lot of settings plus different information in it, and retrieve such info from the registry all of the time. The registry is furthermore a bottleneck in which because it really is the heart of the operating system, any issues with it could cause mistakes and bring the operating system down.<br><br>Registry cleaners have been tailored for 1 purpose - to wash out the 'registry'. This is the central database which Windows relies on to function. Without this database, Windows wouldn't even exist. It's so significant, which the computer is regularly adding plus updating the files inside it, even when you're browsing the Internet (like now). This really is great, nevertheless the problems occur whenever certain of those files become corrupt or lost. This happens a lot, and it takes a good tool to fix it.<br><br>Google Chrome crashes on Windows 7 when the registry entries are improperly modified. Missing registry keys or registry keys with improper values will lead to runtime errors plus thereby the problem happens. You are suggested to scan the entire program registry and review the outcome. Attempt the registry repair task utilizing third-party [http://bestregistrycleanerfix.com/registry-mechanic registry mechanic] software.<br><br>If you think which there are issues with the d3d9.dll file, then we have to substitute it with a unique working file. This may be performed by conducting a series of steps plus we can start by getting "d3d9.zip" within the server. Then you need to unzip the "d3d9.dll" file on the hard drive of the computer. Proceed by finding "C:\Windows\System32" and then finding the existing "d3d9.dll" on your PC. When found, rename the file "d3d9.dll to d3d9BACKUP.dll" plus then copy-paste this brand-new file to "C:\Windows\System32". After that, press "Start" followed by "Run" or search "Run" on Windows Vista & 7. As soon because a box shows up, kind "cmd". A black screen can then appear and you need to kind "regsvr32d3d9.dll" and then click "Enter". This task may enable we to replace the older file with the fresh copy.<br><br>Your disk demands area in order to run smoothly. By freeing up certain space from your disk, you'll be able to speed up the PC a bit. Delete all file in the temporary internet files folder, recycle bin, clear shortcuts plus icons from your desktop which we do not employ and remove programs you do not employ.<br><br>So, the greatest thing to do whenever a computer runs slow is to buy an authentic and legal registry repair tool which would assist you eliminate all difficulties connected to registry plus help you enjoy a smooth running computer. |
| [[File:Triaxial Ellipsoid.jpg|thumb|Tri-axial ellipsoid with distinct semi-axes lengths <math>c>b>a</math>]]
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| [[File:Ellipsoid tri-axial abc.svg|thumb|200px|Tri-axial ellipsoid with distinct semi-axes ''a'', ''b'' and ''c'']]
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| [[File:Ellipsoid revolution prolate and oblate aac.svg|thumb|200px|Ellipsoids of revolution ([[spheroid]]) with a pair of equal semi-axes (''a'') and a distinct third semi-axis (''c'') which is an axis of symmetry. The ellipsoid is oblate or prolate as ''c'' is less than or greater than ''a''.]]
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| An '''ellipsoid''' is a closed [[Quadric|quadric surface]] that is a three-dimensional analogue of an [[ellipse]]. The standard equation of an ellipsoid centered at the origin of a [[Cartesian coordinate system]] and aligned with the axes is
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| :<math>{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1,</math>
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| 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 '''semi-principal axes''' of length ''a'', ''b'', ''c''. They correspond to the [[semi-major axis]] and [[semi-minor axis]] of the appropriate [[ellipse]]s.
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| There are four distinct cases of which one is degenerate: | |
| *<math>a>b>c</math> — '''tri-axial''' or (rarely) '''scalene''' ellipsoid;
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| *<math>a=b>c</math> — '''oblate''' ellipsoid of revolution ([[oblate spheroid]]);
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| *<math>a=b<c</math> — '''prolate''' ellipsoid of revolution ([[prolate spheroid]]);
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| *<math>a=b=c</math> — the '''degenerate''' case of a '''[[sphere]]''';
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| Mathematical literature often uses 'ellipsoid' in place of 'tri-axial ellipsoid'. Scientific literature (particularly geodesy) often uses 'ellipsoid' in place of 'ellipsoid of revolution' and only applies the adjective 'tri-axial' when treating the general case. Older literature uses '[[spheroid]]' in place of 'ellipsoid of revolution'.
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| Any planar [[Cross section (geometry)|cross section]] passing through the center of an ellipsoid forms an ellipse on its surface: this degenerates to a circle for sections normal to the symmetry axis of an ellipsoid of revolution (or all sections when the ellipsoid degenerates to a sphere.)
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| ==Generalised equations==
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| More generally, an arbitrarily oriented ellipsoid, centered at '''v''', is defined by the solutions '''x''' to the equation
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| :<math>(\mathbf{x-v})^\mathrm{T}\! A\, (\mathbf{x-v}) = 1,</math>
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| where ''A'' is a [[positive definite matrix]] and '''x''', '''v''' are [[euclidean vector|vectors]].
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| 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>http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf</ref>
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| 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 semiaxes are given by the eigenvalues. The [[singular value decomposition]] and [[polar decomposition]] are matrix decompositions closely related to these geometric observations.
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| == Parameterization ==
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| The surface of the ellipsoid may be parameterized in several ways. One possible choice which singles out the 'z'-axis is:
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| ::<math>\begin{align}
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| x&=a\,\cos u\cos v,\\
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| y&=b\,\cos u\sin v,\\
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| z&=c\,\sin u;\end{align}\,\!</math>
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| ::where
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| ::::<math>
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| -{\pi}/{2}\leq u\leq+{\pi}/{2}, | |
| \qquad
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| -\pi\leq v\leq+\pi.\!\,\!
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| </math>
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| The parameters may be interpreted as [[Spherical coordinate system|spherical coordinates]]. For constant ''u'', that is on the ellipse which is the intercept with a constant ''z'' plane, ''v'' then plays the role of the [[eccentric anomaly]] for that ellipse. For constant ''v'' on a plane through the ''Oz'' axis the parameter ''u'' plays the same role for the ellipse of intersection. Two other similar parameterizations are possible, each with their own interpretations. Only on an ellipse of revolution can a unique definition of [[Latitude#Reduced (or parametric) latitude|reduced latitude]] be made.
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| ==Volume and surface area==
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| ===Volume===
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| The [[volume]] of an ellipsoid is given by the formula
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| ::<math> V= \frac{4}{3}\pi abc = \frac{4}{3}\pi\sqrt{\det(A^{-1})} .\,\!</math>
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| 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.
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| The [[volumes]] of the maximum [[inscribed]] and minimum [[circumscribed]] [[rectangular cuboid|boxes]] are respectively:
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| ::<math>V_\max =\frac{8}{3\sqrt 3} abc, \qquad V_\min = 8abc.</math>
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| The volume of an ellipse of dimension higher than 3 can be calculated using the dimensional constant given for the [[Hypersphere#Volume and surface area|volume of a hypersphere]].
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| 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.
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| ===Surface area===
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| 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 http://dlmf.nist.gov/19.33 (see next reference).</ref><ref>NIST (National Institute of Standards and Technology) athttp://www.nist.gov</ref>
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| ::<math>
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| S=2\pi c^2+ \frac{2\pi ab}{\sin\phi}
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| \left(E(\phi,k)\, \sin^2\phi + F(\phi,k)\, \cos^2\phi \right),
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| </math>
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| ::where
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| ::<math>
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| \cos\phi = \frac{c}{a}, \qquad
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| k^2 =\frac{a^2(b^2-c^2)}{b^2(a^2-c^2)}, \qquad
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| a\ge b \ge c,
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| </math>
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| and F(φ,k), E(φ,k) are incomplete [[elliptic integral]]s of the first and second kind respectively.[http://dlmf.nist.gov/19.2]
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| The surface area of an ellipsoid of revolution (or [[spheroid]]) may be expressed in terms of [[elementary function]]s:
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| ::<math>S_{\rm oblate} = 2\pi a^2\left(1+\frac{1-e^2}{e}\tanh^{-1}e\right)
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| \quad\mbox{where}\quad e^2=1-\frac{c^2}{a^2}\quad(c<a), </math>
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| ::<math>S_{\rm prolate} = 2\pi a^2\left(1+\frac{c}{ae}\sin^{-1}e\right)
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| \quad\qquad\mbox{where}\;\quad e^2=1-\frac{a^2}{c^2}\quad(c>a), </math>
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| 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 at [http://mathworld.wolfram.com/ProlateSpheroid.html Mathworld]. | |
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| ====Approximate formula====
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| :: <math>S\approx 4\pi\!\left(\frac{ a^p b^p+a^p c^p+b^p c^p }{3}\right)^{1/p}.\,\!</math> | |
| Here ''p'' ≈ 1.6075 yields a relative error of at most 1.061%;<ref>[http://www.numericana.com/answer/ellipsoid.htm#thomsen Final answers] by Gerard P. Michon (2004-05-13). See Thomsen's formulas and Cantrell's comments.</ref> a value of ''p'' = 8/5 = 1.6 is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.
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| In the "flat" limit of ''c'' much smaller than ''a, b'', the area is approximately 2πab.
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| ==Dynamical properties==
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| The [[mass]] of an ellipsoid of uniform density ρ is:
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| :<math>m = \rho V = \rho \frac{4}{3} \pi abc\,\!</math> | |
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| The [[Moment of Inertia|moments of inertia]] of an ellipsoid of uniform density are:
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| ::<math>I_{\mathrm{xx}} = \frac{1}{5} m( b^2+c^2),\qquad
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| I_{\mathrm{yy}} = \frac{1}{5} m(c^2+a^2),\qquad
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| I_{\mathrm{zz}} = \frac{1}{5} m(a^2+b^2),</math>
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| ::<math> I_{\mathrm{xy}}= I_{\mathrm{yz}} = I_{\mathrm{zx}} =0.\,\!</math>
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| For a=b=c these moments of inertia reduce to those for a sphere of uniform density.
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| [[File:2003EL61art.jpg|right|thumb|Artist's conception of {{dp|Haumea}}, a Jacobi-ellipsoid [[dwarf planet]], with its two moons]]
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| 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>
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| One practical effect of this is that scalene astronomical bodies such as {{dp|Haumea}} generally rotate along their minor axes (as does the 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.
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| A relaxed ellipsoid, that is, one in [[hydrostatic equilibrium]], has an oblateness {{nowrap|''a − c''}} directly proportional to its mean density and mean radius. Ellipsoids with a differentiated interior—that is, a denser core than mantle—have a lower oblateness than a homogeneous body. Over all, the ratio (''b–c'')/(''a−c'') is approximately 0.25, though this drops for rapidly rotating bodies.<ref>{{cite web|title=Shapes of the Saturnian Icy Satellites|url=http://www.lpi.usra.edu/meetings/lpsc2006/pdf/1639.pdf}}</ref>
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| The terminology typically used for bodies rotating on their minor axis and whose shape is determined by their gravitational field is '''''Maclaurin spheroid''''' (oblate spheroid) and '''''Jacobi ellipsoid''''' (scalene ellipsoid). At faster rotations, piriform or oviform shapes can be expected, but these are not stable.
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| == Fluid properties ==
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| 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, Mass. ISBN 978-0-674-03116-6.</ref>
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| == See also ==
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| * [[Paraboloid]]
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| * [[Poinsot's ellipsoid]]
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| * [[Hyperboloid]]
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| * [[Reference ellipsoid]]
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| * [[Geoid]]
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| * [[Ellipsoid method]]
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| * [[Superellipse|Superellipsoid]]
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| * {{dp|Haumea}}, a scalene-ellipsoid-shaped dwarf planet
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| * [[Homoeoid]], a shell bounded by two concentric, similar ellipsoids
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| * [[Focaloid]], a shell bounded by two concentric, confocal ellipsoids
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| * [[Elliptical distribution]], in statistics
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| * [[Ellipse]]
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| == References ==
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| <references />
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| * "[http://demonstrations.wolfram.com/Ellipsoid/ Ellipsoid]" by Jeff Bryant, [[Wolfram Demonstrations Project]], 2007.
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| * [http://mathworld.wolfram.com/Ellipsoid.html Ellipsoid] and [http://mathworld.wolfram.com/QuadraticSurface.html Quadratic Surface], [[MathWorld]].
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| == External links ==
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| {{commons category|Ellipsoids}}
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| * [http://www.start2code.com/Cresources/ellipsoid-program-cpp.html Program in C++ to draw an ellipsoid using parametric equation]
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| * [http://code.google.com/p/ellipsoids Ellipsoidal Toolbox for MATLAB]
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| [[Category:Geometric shapes]]
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| [[Category:Surfaces]]
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| [[Category:Quadrics]]
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| [[ta:நீளுருண்டை]]
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