Ellipsoid_(geometry)

Ellipsoid (geometry)

This article is about the shape. For the type of theatrical spotlight, see ellipsoidal reflector spotlight.

Shaded wireframe rendering of an ellipsoid with a = 3, b = 2, c = 1 (scalene ellipsoid).

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 xyz-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 (along the x and y axes) and c is the polar radius (along the z-axis), 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:

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

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. These correspond to the semi-major axis and semi-minor axis of the appropriate ellipses.

1 Parameterization
2 Volume
3 Surface area
4 Mass properties
5 Linear transformations
6 Egg shape
7 References
8 See also
9 External links

Parameterization

Using the common coordinates, where $\beta\,\!$ is a point's reduced, or parametric latitude and ${\color{white}+}\!\!\!\lambda{\color{white}'}\,\!$ is its planetographic longitude, an ellipsoid can be parameterized by:

$\begin{align} x&=a\,\cos(\beta)\cos(\lambda);\!{\color{white}|}\\ y&=b\,\cos(\beta)\sin(\lambda);\\ z&=c\,\sin(\beta);\end{align}\,\!$

$\begin{matrix}-90^\circ\leq\beta\leq+90^\circ; \quad-180^\circ\leq\lambda\leq+180^\circ;\!{\color{white}\big|}\end{matrix}\,\!$

(Note that this parameterization is not

1-1 at the poles, where $\scriptstyle{{\color{white}|}\beta=\pm{90}^\circ}{\color{white}|}\,\!$ )

Or, using spherical coordinates, where ${\color{white}+}\!\!\!\theta{\color{white}'}\,\!$ is the colatitude, or zenith, and ${\color{white}+}\!\!\!\varphi{\color{white}\!\!\!-}\,\!$ is the longitude in 360°, or azimuth:

$\begin{align} x&=a\,\sin(\theta)\cos(\varphi);\!{\color{white}|}\\ y&=b\,\sin(\theta)\sin(\varphi);\\ z&=c\,\cos(\theta);\end{align}\,\!$

$\begin{matrix}0\leq\theta\leq{180}^\circ; \quad{0}\leq\varphi\leq{360}^\circ;\!{\color{white}\big|}\end{matrix}\,\!$

Volume

The volume of an ellipsoid is given by the formula

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

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 or prolate spheroid when two of them are equal.

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.

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:

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

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

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:

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

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-density 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:

$a = \sqrt{{5 \over 2} {I_{\mathrm{yy}}+I_{\mathrm{zz}}-I_{\mathrm{xx}} \over m}}$

$b = \sqrt{{5 \over 2} {I_{\mathrm{zz}}+I_{\mathrm{xx}}-I_{\mathrm{yy}} \over m}}$

$c = \sqrt{{5 \over 2} {I_{\mathrm{xx}}+I_{\mathrm{yy}}-I_{\mathrm{zz}} \over m}}$

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

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

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.^[1] 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).

References

^ Egg Curves by Jürgen Köller.

"Ellipsoid" by Jeff Bryant, The Wolfram Demonstrations Project, 2007.
Ellipsoid and Quadratic Surface, MathWorld.

External links

Interactive Java 3D model of the ellipsoid

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