Ellipsoids

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 x-y-z Cartesian coordinate system is

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

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:

• $a=b=c:\backslash ,\backslash !$ Sphere;
• $a=b>c:\backslash ,\backslash !$ Oblate spheroid (disk-shaped);
• Prolate spheroid (cigar-shaped);
• $a>b>c:\backslash ,\backslash !$ 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.

Contents 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 $\backslash beta\backslash ,\backslash !$ is a point\'s reduced, or parametric latitude and $\{\backslash color\{white\}+\}\backslash !\backslash !\backslash !\backslash lambda\{\backslash color\{white\}\backslash \text{'}\}\backslash ,\backslash !$ is its planetographic longitude, an ellipsoid can be parameterized by:

$\backslash begin\{align\}$

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

$\backslash begin\{matrix\}-90^\backslash circ\backslash leq\backslash beta\backslash leq+90^\backslash circ;$

\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 $\backslash scriptstyle\{\{\backslash color\{white\}|\}\backslash beta=\backslash pm\{90\}^\backslash circ\}\{\backslash color\{white\}|\}\backslash ,\backslash !$)

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

$\backslash begin\{align\}$

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

$\backslash begin\{matrix\}0\backslash leq\backslash theta\backslash leq\{180\}^\backslash circ;$

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

Volume

The volume of an ellipsoid is given by the formula

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

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:

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

where

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

is the modular angle, or angular eccentricity; $m=\backslash frac\{b^2-c^2\}\{b^2\backslash sin(o\backslash !\backslash varepsilon)^2\}\backslash ,\backslash !$ and $E(o\backslash !\backslash varepsilon,m)\backslash ,\backslash !$, $F(o\backslash !\backslash varepsilon,m)\backslash ,\backslash !$ 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:

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

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

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

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

Mass properties

The mass of an ellipsoid of uniform density is:

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

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

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

$I\_\{\backslash mathrm\{xx\}\}\; =\; m\; \{b^2+c^2\; \backslash over\; 5\}$

$I\_\{\backslash mathrm\{yy\}\}\; =\; m\; \{c^2+a^2\; \backslash over\; 5\}$

$I\_\{\backslash mathrm\{zz\}\}\; =\; m\; \{a^2+b^2\; \backslash over\; 5\}$

where $I\_\{\backslash mathrm\{xx\}\}\backslash ,\backslash !$, $I\_\{\backslash mathrm\{yy\}\}\backslash ,\backslash !$, and $I\_\{\backslash mathrm\{zz\}\}\backslash ,\backslash !$ 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:

$a\; =\; \backslash sqrt\{\{5\; \backslash over\; 2\}\; \{I\_\{\backslash mathrm\{yy\}\}+I\_\{\backslash mathrm\{zz\}\}-I\_\{\backslash mathrm\{xx\}\}\; \backslash over\; m\}\}$

$b\; =\; \backslash sqrt\{\{5\; \backslash over\; 2\}\; \{I\_\{\backslash mathrm\{zz\}\}+I\_\{\backslash mathrm\{xx\}\}-I\_\{\backslash mathrm\{yy\}\}\; \backslash over\; m\}\}$

$c\; =\; \backslash sqrt\{\{5\; \backslash over\; 2\}\; \{I\_\{\backslash mathrm\{xx\}\}+I\_\{\backslash mathrm\{yy\}\}-I\_\{\backslash mathrm\{zz\}\}\; \backslash over\; m\}\}$

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

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. 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

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

See also

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

External links

• Interactive Java 3D model of the ellipsoid

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