Ellipsoid

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface;  that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is Bounded set, which means that it may be enclosed in a sufficiently large sphere.

An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a Central symmetry, called the center of the ellipsoid. The that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a triaxial ellipsoid (rarely scalene ellipsoid), and the axes are uniquely defined.

If two of the axes have the same length, then the ellipsoid is an ellipsoid of revolution, also called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, and there are thus infinitely many ways of choosing the two perpendicular axes of the same length. In the case of two axes being the same length:

• If the third axis is shorter, the ellipsoid is a sphere that has been flattened (called an oblate spheroid).
• If the third axis is longer, it is a sphere that has been lengthened (called a prolate spheroid).

If the three axes have the same length, the ellipsoid is a sphere.

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

The points $(a,\; 0,\; 0)$, $(0,\; b,\; 0)$ and $(0,\; 0,\; c)$ lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because are half the length of the principal axes. They correspond to the semi-major axis and semi-minor axis of an ellipse.

In spherical coordinate system for which $(x,y,z)=(r\backslash sin\backslash theta\backslash cos\backslash varphi,\; r\backslash sin\backslash theta\backslash sin\backslash varphi,r\backslash cos\backslash theta)$, the general ellipsoid is defined as:

$\{r^2\backslash sin^2\backslash theta\backslash cos^2\backslash varphi\backslash over\; a^2\}+\{r^2\backslash sin^2\backslash theta\backslash sin^2\backslash varphi\; \backslash over\; b^2\}+\{r^2\backslash cos^2\backslash theta\; \backslash over\; c^2\}=1,$

where $\backslash theta$ is the polar angle and $\backslash varphi$ is the azimuthal angle.

When $a=b=c$, the ellipsoid is a sphere.

When $a=b\backslash neq\; c$, the ellipsoid is a spheroid or ellipsoid of revolution. In particular, if $a\; =\; b\; >\; c$, it is an oblate spheroid; if , it is a prolate spheroid.

$\backslash begin\{align\}$

 x &= a\sin\theta\cos\varphi,\\
 y &= b\sin\theta\sin\varphi,\\
 z &= c\cos\theta,
     

where

 0 \le \theta \le \pi,\qquad
 0 \le \varphi < 2\pi.
     

These parameters may be interpreted as spherical coordinates, where is the polar angle and is the azimuth angle of the point of the ellipsoid..

Measuring from the equator rather than a pole,

$\backslash begin\{align\}$

 x &= a\cos\theta\cos\lambda,\\
 y &= b\cos\theta\sin\lambda,\\
 z &= c\sin\theta,
     

where

 -\tfrac{\pi}2 \le \theta \le \tfrac{\pi}2,\qquad
 0 \le \lambda < 2\pi,
     

is the [[reduced latitude]], parametric latitude, or eccentric anomaly and  is azimuth or longitude.
     

Measuring angles directly to the surface of the ellipsoid, not to the circumscribed sphere,

$\backslash begin\{bmatrix\}$

   x \\ y \\ z
 \end{bmatrix} =
 R \begin{bmatrix}
   \cos\gamma\cos\lambda\\
   \cos\gamma\sin\lambda\\
   \sin\gamma
 \end{bmatrix}
     

where

$\backslash begin\{align\}$

 R ={} &\frac{abc}{\sqrt{c^2 \left(b^2\cos^2\lambda + a^2\sin^2\lambda\right) \cos^2\gamma
              + a^2 b^2\sin^2\gamma}}, \\[3pt]
       &-\tfrac{\pi}2 \le \gamma \le \tfrac{\pi}2,\qquad
          0 \le \lambda < 2\pi.
     

would be geocentric latitude on the Earth, and  is longitude. These are true spherical coordinates with the origin at the center of the ellipsoid.
     

In geodesy, the geodetic latitude is most commonly used, as the angle between the vertical and the equatorial plane, defined for a biaxial ellipsoid. For a more general triaxial ellipsoid, see ellipsoidal latitude.

$V\; =\; \backslash tfrac\{4\}\{3\}\backslash pi\; abc.$

$V\; =\; \backslash tfrac16\; \backslash pi\; ABC$.

The volume of an ellipsoid is the volume of a circumscribed elliptic cylinder, and the volume of the circumscribed box. The volumes of the inscribed and circumscribed boxes are respectively:

 V_\text{inscribed} = \frac{8}{3\sqrt{3}} abc,\qquad
 V_\text{circumscribed} = 8abc.
     

$S\; =\; 2\backslash pi\; c^2\; +\; \backslash frac\{2\backslash pi\; ab\}\{\backslash sin(\backslash varphi)\}\backslash left(E(\backslash varphi,\; k)\backslash ,\backslash sin^2(\backslash varphi)\; +\; F(\backslash varphi,\; k)\backslash ,\backslash cos^2(\backslash varphi)\backslash right),$

where

 \cos(\varphi) = \frac{c}{a},\qquad
 k^2 = \frac{a^2\left(b^2 - c^2\right)}{b^2\left(a^2 - c^2\right)},\qquad
 a \ge b \ge c,
     

and where and are incomplete elliptic integrals of the first and second kind respectively.

The surface area of this general ellipsoid can also be expressed in terms of , one of the Carlson symmetric forms of elliptic integrals:

$S\; =\; 4\backslash pi\; bc\; R\_\{G\}\; \backslash left(\; \backslash frac\{a^2\}\{b^2\}\; ,\; \backslash frac\{a^2\}\{c^2\}\; ,\; 1\backslash right).$

Simplifying the above formula using properties of , this can also be expressed in terms of the volume of the ellipsoid :

$S\; =\; 3VR\_\{G\}\backslash left(a^\{-2\},b^\{-2\},c^\{-2\}\backslash right).$

Unlike the expression with and , the equations in terms of do not depend on the choice of an order on , , and .

The surface area of an ellipsoid of revolution (or spheroid) may be expressed in terms of elementary functions:

  S_\text{oblate} = 2\pi a^2\left(1 + \frac{c^2}{ea^2} \operatorname{artanh}e\right),
                  \qquad\text{where }e^2 = 1 - \frac{c^2}{a^2}\text{ and }(c < a), 
     

S_\text{oblate} = 2\pi a^2\left(1 + \frac{1 - e^2}{e} \operatorname{artanh}e\right)
     

or

and

 S_\text{prolate} = 2\pi a^2\left(1 + \frac{c}{ae}  \arcsin e\right)
                  \qquad\text{where } e^2 = 1 - \frac{a^2}{c^2}\text{ and } (c > a),
     

which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases may again be identified as the eccentricity of the ellipse formed by the cross section through the symmetry axis. (See ellipse). Derivations of these results may be found in standard sources, for example Mathworld.

$S\; \backslash approx\; 4\backslash pi\; \backslash sqrtp\{\backslash frac\{a^p\; b^p\; +\; a^p\; c^p\; +\; b^p\; c^p\}\{3\}\}.\backslash ,\backslash !$

Here yields a relative error of at most 1.061%; Final answers by Gerard P. Michon (2004-05-13). See Thomsen's formulas and Cantrell's comments. a value of is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.

In the "flat" limit of much smaller than and , the area is approximately , equivalent to .

Wanted: Three vectors (center) and , (conjugate vectors), such that the ellipse can be represented by the parametric equation

$\backslash mathbf\; x\; =\; \backslash mathbf\; f\_0\; +\; \backslash mathbf\; f\_1\backslash cos\; t\; +\; \backslash mathbf\; f\_2\backslash sin\; t$

Solution: The scaling transforms the ellipsoid onto the unit sphere and the given plane onto the plane with equation

$\backslash \; n\_x\; au\; +\; n\_y\; bv\; +\; n\_z\; cw\; =\; d.$

$\backslash ;\backslash mathbf\; m\; =\; \backslash begin\{bmatrix\}\; m\_u\; \backslash \backslash \; m\_v\; \backslash \backslash \; m\_w\; \backslash end\{bmatrix\}\backslash ;$

$\backslash mathbf\; e\_0\; =\; \backslash delta\; \backslash mathbf\; m\; \backslash ;$

$\backslash ;\backslash rho\; =\; \backslash sqrt\{1\; -\; \backslash delta^2\}\backslash ;$

Where (i.e. the plane is horizontal), let

$\backslash \; \backslash mathbf\; e\_1\; =\; \backslash begin\{bmatrix\}\; \backslash rho\; \backslash \backslash \; 0\; \backslash \backslash \; 0\; \backslash end\{bmatrix\},\backslash qquad\; \backslash mathbf\; e\_2\; =\; \backslash begin\{bmatrix\}\; 0\; \backslash \backslash \; \backslash rho\; \backslash \backslash \; 0\; \backslash end\{bmatrix\}.$

Where , let

$\backslash mathbf\; e\_1\; =\; \backslash frac\{\backslash rho\}\{\backslash sqrt\{m\_u^2\; +\; m\_v^2\}\}\backslash ,\; \backslash begin\{bmatrix\}\; m\_v\; \backslash \backslash \; -m\_u\; \backslash \backslash \; 0\; \backslash end\{bmatrix\}\backslash ,\; ,\backslash qquad\; \backslash mathbf\; e\_2\; =\; \backslash mathbf\; m\; \backslash times\; \backslash mathbf\; e\_1\backslash \; .$

In any case, the vectors are orthogonal, parallel to the intersection plane and have length (radius of the circle). Hence the intersection circle can be described by the parametric equation

$\backslash ;\backslash mathbf\; u\; =\; \backslash mathbf\; e\_0\; +\; \backslash mathbf\; e\_1\backslash cos\; t\; +\; \backslash mathbf\; e\_2\backslash sin\; t\backslash ;.$

The reverse scaling (see above) transforms the unit sphere back to the ellipsoid and the vectors are mapped onto vectors , which were wanted for the parametric representation of the intersection ellipse.

How to find the vertices and semi-axes of the ellipse is described in ellipse.

Example: The diagrams show an ellipsoid with the semi-axes which is cut by the plane .

A pins-and-string construction of an ellipsoid of revolution is given by the pins-and-string construction of the rotated ellipse.

The construction of points of a triaxial ellipsoid is more complicated. First ideas are due to the Scottish physicist J. C. Maxwell (1868). W. Böhm: Die FadenKonstruktion der Flächen zweiter Ordnung, Mathemat. Nachrichten 13, 1955, S. 151 Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898.Staude, O.: Ueber Fadenconstructionen des Ellipsoides. Math. Ann. 20, 147–184 (1882) Staude, O.: Ueber neue Focaleigenschaften der Flächen 2. Grades. Math. Ann. 27, 253–271 (1886). Staude, O.: Die algebraischen Grundlagen der Focaleigenschaften der Flächen 2. Ordnung Math. Ann. 50, 398 - 428 (1898). A description of the pins-and-string construction of ellipsoids and hyperboloids is contained in the book Geometry and the Imagination by David Hilbert & Cohn-Vossen.D. Hilbert & S Cohn-Vossen: Geometry and the imagination, Chelsea New York, 1952, , p. 20

1. Choose an ellipse and a hyperbola , which are a pair of focal conics: $$\backslash begin\{align\}$$

1. Pin one end of the string to vertex and the other to focus . The string is kept tight at a point with positive - and -coordinates, such that the string runs from to behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from to runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance over any hyperbola point is at a minimum. The analogous statement on the second part of the string and the ellipse has to be true, too.
2. Then: is a point of the ellipsoid with equation $$\backslash begin\{align\}$$

&\frac{x^2}{r_x^2} + \frac{y^2}{r_y^2} + \frac{z^2}{r_z^2} = 1 \\
&r_x = \tfrac{1}{2}(l - a + c), \quad
 r_y = {\textstyle \sqrt{r^2_x - c^2}}, \quad
 r_z = {\textstyle \sqrt{r^2_x - a^2}}.
     

1. The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.

$Y\; =\; (0,\; r\_y,\; 0),\backslash quad\; Z\; =\; (0,\; 0,\; r\_z).$

The lower part of the diagram shows that and are the foci of the ellipse in the -plane, too. Hence, it is confocal to the given ellipse and the length of the string is . Solving for yields ; furthermore .

From the upper diagram we see that and are the foci of the ellipse section of the ellipsoid in the -plane and that .

$\backslash overline\; r\_x^2\; =\; r\_x^2\; -\; \backslash lambda,\; \backslash quad$

 \overline r_y^2 = r_y^2 - \lambda, \quad
 \overline r_z^2 = r_z^2 - \lambda
     

then from the equations of

$r\_x^2\; -\; r\_y^2\; =\; c^2,\; \backslash quad$

 r_x^2 - r_z^2 = a^2, \quad
 r_y^2 - r_z^2 = a^2 - c^2 = b^2
     

one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes as ellipsoid . Therefore (analogously to the foci of an ellipse) one considers the focal conics of a triaxial ellipsoid as the (infinite many) foci and calls them the focal curves of the ellipsoid.O. Hesse: Analytische Geometrie des Raumes, Teubner, Leipzig 1861, p. 287

The converse statement is true, too: if one chooses a second string of length and defines

$\backslash lambda\; =\; r^2\_x\; -\; \backslash overline\; r^2\_x$

$\backslash overline\; r\_y^2\; =\; r\_y^2\; -\; \backslash lambda,\backslash quad\; \backslash overline\; r\_z^2\; =\; r\_z^2\; -\; \backslash lambda$

$r\_x\; =\; \backslash tfrac12l,\backslash quad\; r\_y\; =\; r\_z\; =\; \{\backslash textstyle\; \backslash sqrt\{r^2\_x\; -\; c^2\}\}$.

True curve

If one views an ellipsoid from an external point of its focal hyperbola, then it seems to be a sphere, that is its apparent shape is a circle. Equivalently, the tangents of the ellipsoid containing point are the lines of a circular cone, whose axis of rotation is the tangent line of the hyperbola at .D. Hilbert & S Cohn-Vossen: Geometry and the Imagination, p. 24O. Hesse: Analytische Geometrie des Raumes, p. 301 If one allows the center to disappear into infinity, one gets an orthogonal parallel projection with the corresponding asymptote of the focal hyperbola as its direction. The true curve of shape (tangent points) on the ellipsoid is not a circle. The lower part of the diagram shows on the left a parallel projection of an ellipsoid (with semi-axes 60, 40, 30) along an asymptote and on the right a central projection with center and main point on the tangent of the hyperbola at point . ( is the foot of the perpendicular from onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin is the circle's center; in the central case main point is the center.

Umbilical points

The focal hyperbola intersects the ellipsoid at its four .W. Blaschke: Analytische Geometrie, p. 125

$r\_x\; =\; a,\backslash quad\; r\_y\; =\; b,\backslash quad\; l\; =\; 3a\; -\; c.$

One can also define a hyperellipsoid as the image of a sphere under an invertible affine transformation. The spectral theorem can again be used to obtain a standard equation of the form

$\backslash frac\{x\_1^2\}\{a\_1^2\}+\backslash frac\{x\_2^2\}\{a\_2^2\}+\backslash cdots\; +\; \backslash frac\{x\_n^2\}\{a\_n^2\}=1.$

The volume of an -dimensional hyperellipsoid can be obtained by replacing by the product of the semi-axes in the formula for the volume of a hypersphere:

$V\; =\; \backslash frac\{\backslash pi^\backslash frac\{n\}\{2\}\}\{\backslash Gamma\{\backslash left(\backslash frac\{n\}\{2\}\; +\; 1\backslash right)\}\}\; a\_1a\_2\backslash cdots\; a\_n\; \backslash approx\; \backslash frac\{1\}\{\backslash sqrt\{\backslash pi\; n\}\}\; \backslash cdot\; \backslash left(\backslash frac\{2\; e\; \backslash pi\}\{n\}\backslash right)^\{n/2\}\; a\_1a\_2\backslash cdots\; a\_n$

$(\backslash mathbf\{x\}-\backslash mathbf\{v\})^\backslash mathsf\{T\}\backslash !\; \backslash boldsymbol\{A\}\backslash ,\; (\backslash mathbf\{x\}-\backslash mathbf\{v\})\; =\; 1$

 I_\mathrm{xx} &= \tfrac{1}{5}m\left(b^2 + c^2\right), &
   I_\mathrm{yy} &= \tfrac{1}{5}m\left(c^2 + a^2\right), &
   I_\mathrm{zz} &= \tfrac{1}{5}m\left(a^2 + b^2\right), \\[3pt]
 I_\mathrm{xy} &=  I_\mathrm{yz} = I_\mathrm{zx} = 0.