Ellipsoid

3D rendering of an ellipsoid

In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. The equation of a standard ellipsoid in an x-y-z Cartesian coordinate system is

[++=1]

(where a, b and c (the lengths of the three semi-axes) are fixed positive real numbers determining the shape of the ellipsoid. If two of those numbers are equal, the ellipsoid is a spheroid; if all three are equal, it is a sphere.

If it is assumed a ≥ b ≥ c, then, when:

a = b = c, it is the aforementioned sphere;
b = c, a > b, the ellipsoid is an oblate spheroid (disk-shaped);
b = c, a < b, the ellipsoid is a prolate spheroid (cigar-shaped);
a ≠ b ≠ c, the ellipsoid is scalene.

Contents

1 Parameterization
2 Volume
3 Surface area
4 Linear transformations
5 Egg shape
6 See also

Parameterization

An ellipsoid can be parameterized by:

[x=a\,\sin(\phi)\cos(O\!\!E);\,\!]

[y=b\,\sin(\phi)\sin(O\!\!E);\,\!]

[z=c\,\cos(\phi);\,\!]

[0\leq O\!\!E <2\pi;\,\!]

[0\leq\phi\leq\pi,\,\!]

(Note that this parameterization is not 1-1 at the points where [\phi=0,\pi\,\!].)

where

[O\!\!E=\arccos\left(\frac\right)\;\textrm\arccos\left(\frac\right)\;(\textrm),\,\!]

is the modular angle, or angular eccentricity.

Volume

The volume of an ellipsoid is given by:

[\frac\pi abc.\,\!]

Surface area

The surface area of an ellipsoid is given by:

[2 \pi \left( c^2 + \frac} F(O\!\!E,m) + b\sqrt E(O\!\!E,m) \right),\,\!]

where [m=\frac\,\!] and [F(O\!\!E,m)\,\!], [E(O\!\!E,m)\,\!] are the incomplete elliptic integrals of the first and second kind.

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 b = c:

If oblate: [2\pi\!\left(a^2+b^2\frac(\sin(O\!\!E))}\right);\,\!]

If prolate: [2\pi\!\left(\frac(2O\!\!E)}+b^2\right)=2\pi\!\left(a^2\frac+b^2\right);\,\!]

In the "flat" limit of [c \ll a, b], the area is approximately

: [2\pi\!\left(ab\right)]

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 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 intersection of an ellipsoid with a plane is empty, a single point or an ellipse.

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 an egg is approximately an oblate ellipsoid, but, while keeping cylindrical symmetry, there is not quite symmetry in a plane perpendicular to the long axis. The term egg-shaped is typically used taking this asymmetry into account, but it may also simply mean oblate ellipsoid. It can also be used for a 2D shape. See also oval (geometry).