Ellipsoid

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3D rendering of an ellipsoid (prolate spheroid)

Wireframe rendering of an ellipsoid (oblate spheroid)

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 x-y-z Cartesian coordinate system is

${x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1$

where a and b are the equatorial radii (perpendicular to each other) and c is the polar radius (i.e., the lengths of the three semi-axes), all of which are fixed positive real numbers determining the shape of the ellipsoid.

(This contrasts with standard elliptical assignment, where the equatorial radius/semi-axis is defined as "a" and the polar as "b".)

If all three radii are equal, the solid body is a sphere; if the two equatorial radii are equal, the ellipsoid is a spheroid:

$a=b=c:\,\!$ Sphere;
$a=b>c:\,\!$ Oblate spheroid (disk-shaped);
$a=b<c:\,\!$ Prolate spheroid (cigar-shaped);
$a>b>c:\,\!$ Scalene ellipsoid ("three unequal sides").

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

[edit] Parameterization

Where $\beta\,\!$ is a point's parametric latitude and $\lambda\,\!$ is its planetographic longitude, an ellipsoid can be parameterized by:

$x=a\,\cos(\beta)\cos(\lambda);\,\!$

$y=b\,\cos(\beta)\sin(\lambda);\,\!$

$z=c\,\sin(\beta);\,\!$

$\begin{matrix}-\frac{{\color{white}\dot{{\color{black}\pi}}}}{2}\leq\beta\leq+\frac{\pi}{2};\quad-\pi\leq\lambda\leq+\pi;\end{matrix}\,\!$

(Note that this parameterization is not 1-1 at the points where $\begin{matrix}\beta=\pm\frac{{\color{white}\dot{{\color{black}\pi}}}}{2}\end{matrix}\,\!$ .)

[edit] Volume

The volume of an ellipsoid is given by:

$\frac{4}{3}\pi abc.\,\!$

[edit] Surface area

The surface area of an ellipsoid is given by:

$2\pi\left(c^2+b\sqrt{a^2-c^2}E(o\!\varepsilon,m)+\frac{bc^2}{\sqrt{a^2-c^2}}F(o\!\varepsilon,m)\right),\,\!$

where

$o\!\varepsilon=\arccos\left(\frac{c}{a}\right)\;\textrm{(oblate)\;\;or\;\;}\arccos\left(\frac{a}{c}\right)\;(\textrm{prolate}),\,\!$

is the modular angle, or angular eccentricity; $m=\frac{b^2-c^2}{b^2\sin(o\!\varepsilon)^2}\,\!$ and $E(o\!\varepsilon,m)\,\!$ , $F(o\!\varepsilon,m)\,\!$ are the incomplete elliptic integrals of the first and second kind.

An approximate formula is:

$\approx 4\pi\!\left(\frac{ a^p b^p+a^p c^p+b^p c^p }{3}\right)^{1/p}.\,\!$

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: $2\pi\!\left(a^2+c^2\frac{\operatorname{arctanh}(\sin(o\!\varepsilon))}{\sin(o\!\varepsilon)}\right);\,\!$

If prolate: $2\pi\!\left(\frac{a^2}{\operatorname{sinc}(2o\!\varepsilon)}+c^2\right)=2\pi\!\left(a^2\frac{2o\!\varepsilon}{\sin(2o\!\varepsilon)}+c^2\right);\,\!$

In the "flat" limit of $c \ll a, b\,\!$ , the area is approximately $2\pi ab.\,\!$

[edit] Mass Properties

The mass of an ellipsoid of uniform density is:

$m = \rho V = \rho \frac{4}{3} \pi abc\,\!$

where $\rho\,\!$ is the density.

The mass moments of inertia of an ellipsoid of uniform density are:

$I_{\mathrm{xx}} = m {b^2+c^2 \over 5}$

$I_{\mathrm{yy}} = m {c^2+a^2 \over 5}$

$I_{\mathrm{zz}} = m {a^2+b^2 \over 5}$

where $I_{\mathrm{xx}}\,\!$ , $I_{\mathrm{yy}}\,\!$ , and $I_{\mathrm{zz}}\,\!$ 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 uniform sphere.

[edit] 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.

[edit] Egg shape

Oval

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 egg-shaped 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).

[edit] See also

Spheroid
Paraboloid
Hyperboloid
Reference ellipsoid
Geoid
Ellipsoid method
Superellipsoid
(136108) 2003 EL₆₁, an ellipsoid shaped planetoid

Categories: Surfaces | Quadrics