Ellipsoid

File:Ellipsoide.svg

Examples of ellipsoids with equation Template:Math: Template:Ubl

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, 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 center of symmetry, called the center of the ellipsoid. The line segments 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.

Standard equation

The general ellipsoid, also known as triaxial ellipsoid, is a quadratic surface which is defined in Cartesian coordinates as:

<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,</math>

where <math>a</math>, <math>b</math> and <math>c</math> are the length of the semi-axes.

The points <math>(a, 0, 0)</math>, <math>(0, b, 0)</math> and <math>(0, 0, c)</math> lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because Template:Math 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 <math>(x,y,z)=(r\sin\theta\cos\varphi, r\sin\theta\sin\varphi,r\cos\theta)</math>, the general ellipsoid is defined as:

<math>{r^2\sin^2\theta\cos^2\varphi\over a^2}+{r^2\sin^2\theta\sin^2\varphi \over b^2}+{r^2\cos^2\theta \over c^2}=1,</math>

where <math>\theta</math> is the polar angle and <math>\varphi</math> is the azimuthal angle.

When <math>a=b=c</math>, the ellipsoid is a sphere.

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

Parameterization

The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is

<math>\begin{align}

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

\end{align}\,\!</math>

where

<math>

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

</math>

These parameters may be interpreted as spherical coordinates, where Template:Mvar is the polar angle and Template:Mvar is the azimuth angle of the point Template:Math of the ellipsoid.<ref>Template:Harvtxt</ref>

Measuring from the equator rather than a pole,

<math>\begin{align}

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

\end{align}\,\!</math>

where

<math>

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

</math>

Template:Mvar is the reduced latitude, parametric latitude, or eccentric anomaly and Template:Mvar is azimuth or longitude.

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

<math>\begin{bmatrix}

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

\,\!</math>

where

<math>\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.

\end{align}</math>

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

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.

Volume

The volume bounded by the ellipsoid is

<math>V = \tfrac{4}{3}\pi abc.</math>

In terms of the principal diameters Template:Math (where Template:Math, Template:Math, Template:Math), the volume is

<math>V = \tfrac16 \pi ABC</math>.

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.

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

<math>

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

</math>

Surface area

The surface area of a general (triaxial) ellipsoid is<ref>F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark, editors, 2010, NIST Handbook of Mathematical Functions (Cambridge University Press), Section 19.33 {{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>S = 2\pi c^2 + \frac{2\pi ab}{\sin(\varphi)}\left(E(\varphi, k)\,\sin^2(\varphi) + F(\varphi, k)\,\cos^2(\varphi)\right),</math>

where

<math>

 \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,

</math>

and where Template:Math and Template:Math are incomplete elliptic integrals of the first and second kind respectively.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The surface area of this general ellipsoid can also be expressed in terms of Template:Tmath, one of the Carlson symmetric forms of elliptic integrals:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>S = 4\pi bc R_{G} \left( \frac{a^2}{b^2} , \frac{a^2}{c^2} , 1\right).</math>

Simplifying the above formula using properties of Template:Math,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> this can also be expressed in terms of the volume of the ellipsoid Template:Math:

<math>S = 3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right).</math>

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

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

<math>

  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), </math>

or

<math>

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

or

<math>

S_\text{oblate} = 2\pi a^2\ + \frac{\pi c^2}{e}\ln\frac{1+e}{1-e}</math>

and

<math>

 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),

</math>

which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for Template:Math can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases Template:Mvar 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Approximate formula

<math>S \approx 4\pi \sqrt[p]{\frac{a^p b^p + a^p c^p + b^p c^p}{3}}.\,\!</math>

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

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

Plane sections

File:Ellipsoid-ebener-Schnitt.svg

Plane section of an ellipsoid

The intersection of a plane and a sphere is a circle (or is reduced to a single point, or is empty). Any ellipsoid is the image of the unit sphere under some affine transformation, and any plane is the image of some other plane under the same transformation. So, because affine transformations map circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty.<ref>Template:Citation</ref> Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see Circular section).

Determining the ellipse of a plane section

File:Ellipso-eb-beisp.svg

Plane section of an ellipsoid (see example)

Given: Ellipsoid Template:Math and the plane with equation Template:Math, which have an ellipse in common.

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

<math>\mathbf x = \mathbf f_0 + \mathbf f_1\cos t + \mathbf f_2\sin t</math>

(see ellipse).

File:Ellipso-eb-ku.svg

Plane section of the unit sphere (see example)

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

<math>\ n_x au + n_y bv + n_z cw = d. </math>

Let Template:Math be the Hesse normal form of the new plane and

<math>\;\mathbf m = \begin{bmatrix} m_u \\ m_v \\ m_w \end{bmatrix}\;</math>

its unit normal vector. Hence

<math>\mathbf e_0 = \delta \mathbf m \;</math>

is the center of the intersection circle and

<math>\;\rho = \sqrt{1 - \delta^2}\;</math>

its radius (see diagram).

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

<math>\ \mathbf e_1 = \begin{bmatrix} \rho \\ 0 \\ 0 \end{bmatrix},\qquad \mathbf e_2 = \begin{bmatrix} 0 \\ \rho \\ 0 \end{bmatrix}.</math>

Where Template:Math, let

<math >\mathbf e_1 = \frac{\rho}{\sqrt{m_u^2 + m_v^2}}\, \begin{bmatrix} m_v \\ -m_u \\ 0 \end{bmatrix}\, ,\qquad \mathbf e_2 = \mathbf m \times \mathbf e_1\ .</math>

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

<math>\;\mathbf u = \mathbf e_0 + \mathbf e_1\cos t + \mathbf e_2\sin t\;.</math>

The reverse scaling (see above) transforms the unit sphere back to the ellipsoid and the vectors Template:Math are mapped onto vectors Template:Math, 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 Template:Math which is cut by the plane Template:Math.

Pins-and-string construction

File:Ellipse-gaertner-k.svg

Pins-and-string construction of an ellipse:
Template:Math, length of the string (red)

File:Fokalks-ellipsoid.svg

Pins-and-string construction of an ellipsoid, blue: focal conics

File:Fokalks-ellipsoid-xyz.svg

Determination of the semi axis of the ellipsoid

The pins-and-string construction of an ellipsoid is a transfer of the idea constructing an ellipse using two pins and a string (see diagram).

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).<ref> W. Böhm: Die FadenKonstruktion der Flächen zweiter Ordnung, Mathemat. Nachrichten 13, 1955, S. 151</ref> Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898.<ref>Staude, O.: Ueber Fadenconstructionen des Ellipsoides. Math. Ann. 20, 147–184 (1882)</ref><ref> Staude, O.: Ueber neue Focaleigenschaften der Flächen 2. Grades. Math. Ann. 27, 253–271 (1886).</ref><ref> Staude, O.: Die algebraischen Grundlagen der Focaleigenschaften der Flächen 2. Ordnung Math. Ann. 50, 398 - 428 (1898).</ref> The description of the pins-and-string construction of ellipsoids and hyperboloids is contained in the book Geometry and the imagination written by D. Hilbert & S. Vossen,<ref>D. Hilbert & S Cohn-Vossen: Geometry and the imagination, Chelsea New York, 1952, Template:ISBN, p. 20 .</ref> too.

Steps of the construction

1. Choose an ellipse Template:Mvar and a hyperbola Template:Mvar, which are a pair of focal conics: <math display="block">\begin{align}

E(\varphi) &= (a\cos\varphi, b\sin\varphi, 0) \\ H(\psi) &= (c\cosh\psi, 0, b\sinh\psi),\quad c^2 = a^2 - b^2 \end{align} </math> with the vertices and foci of the ellipse <math display="block">S_1 = (a, 0, 0),\quad F_1 = (c, 0, 0),\quad F_2 = (-c, 0, 0),\quad S_2 = (-a, 0, 0)</math> and a string (in diagram red) of length Template:Mvar.

1. Pin one end of the string to vertex Template:Math and the other to focus Template:Math. The string is kept tight at a point Template:Mvar with positive Template:Mvar- and Template:Mvar-coordinates, such that the string runs from Template:Math to Template:Mvar behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from Template:Mvar to Template:Math runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance Template:Math 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: Template:Mvar is a point of the ellipsoid with equation <math display="block">\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}}.

\end{align}</math>

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

Semi-axes

Equations for the semi-axes of the generated ellipsoid can be derived by special choices for point Template:Mvar:

<math>Y = (0, r_y, 0),\quad Z = (0, 0, r_z).</math>

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

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

Converse

If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters Template:Mvar, Template:Mvar, Template:Mvar for a pins-and-string construction.

Confocal ellipsoids

If Template:Overline is an ellipsoid confocal to Template:Mathcal with the squares of its semi-axes

<math>\overline r_x^2 = r_x^2 - \lambda, \quad

 \overline r_y^2 = r_y^2 - \lambda, \quad
 \overline r_z^2 = r_z^2 - \lambda</math>

then from the equations of Template:Mathcal

<math> r_x^2 - r_y^2 = c^2, \quad

 r_x^2 - r_z^2 = a^2, \quad
 r_y^2 - r_z^2 = a^2 - c^2 = b^2</math>

one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes Template:Math as ellipsoid Template:Mathcal. 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.<ref>O. Hesse: Analytische Geometrie des Raumes, Teubner, Leipzig 1861, p. 287</ref>

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

<math>\lambda = r^2_x - \overline r^2_x</math>

then the equations

<math>\overline r_y^2 = r_y^2 - \lambda,\quad \overline r_z^2 = r_z^2 - \lambda</math>

are valid, which means the two ellipsoids are confocal.

Limit case, ellipsoid of revolution

In case of Template:Math (a spheroid) one gets Template:Math and Template:Math, which means that the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on the Template:Mvar-axis. The ellipsoid is rotationally symmetric around the Template:Mvar-axis and

<math>r_x = \tfrac12l,\quad r_y = r_z = {\textstyle \sqrt{r^2_x - c^2}}</math>.

Properties of the focal hyperbola

File:Ellipsoid-pk-zk.svg

Top: 3-axial Ellipsoid with its focal hyperbola.
Bottom: parallel and central projection of the ellipsoid such that it looks like a sphere, i.e. its apparent shape is a circle

True curve

If one views an ellipsoid from an external point Template:Mvar 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 Template:Mvar are the lines of a circular cone, whose axis of rotation is the tangent line of the hyperbola at Template:Mvar.<ref>D. Hilbert & S Cohn-Vossen: Geometry and the Imagination, p. 24</ref><ref>O. Hesse: Analytische Geometrie des Raumes, p. 301</ref> If one allows the center Template:Mvar 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.Template:Paragraph 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 Template:Mvar and main point Template:Mvar on the tangent of the hyperbola at point Template:Mvar. (Template:Mvar is the foot of the perpendicular from Template:Mvar onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin Template:Mvar is the circle's center; in the central case main point Template:Mvar is the center.

Umbilical points

The focal hyperbola intersects the ellipsoid at its four umbilical points.<ref>W. Blaschke: Analytische Geometrie, p. 125</ref>

Property of the focal ellipse

The focal ellipse together with its inner part can be considered as the limit surface (an infinitely thin ellipsoid) of the pencil of confocal ellipsoids determined by Template:Math for Template:Math. For the limit case one gets

<math>r_x = a,\quad r_y = b,\quad l = 3a - c.</math>

In higher dimensions and general position Template:Anchor

A hyperellipsoid, or ellipsoid of dimension <math>n - 1</math> in a Euclidean space of dimension <math>n</math>, is a quadric hypersurface defined by a polynomial of degree two that has a homogeneous part of degree two which is a positive definite quadratic form.

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

<math>\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+\cdots + \frac{x_n^2}{a_n^2}=1.</math>

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

<math>V = \frac{\pi^\frac{n}{2}}{\Gamma{\left(\frac{n}{2} + 1\right)}} a_1a_2\cdots a_n \approx \frac{1}{\sqrt{\pi n}} \cdot \left(\frac{2 e \pi}{n}\right)^{n/2} a_1a_2\cdots a_n </math>

(where Template:Math is the gamma function).

As a quadric

If Template:Mvar is a real, symmetric, Template:Mvar-by-Template:Mvar positive-definite matrix, and Template:Mvar is a vector in <math>\R^n,</math> then the set of points Template:Math that satisfy the equation

<math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) = 1</math>

is an n-dimensional ellipsoid centered at Template:Mvar. The expression <math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) </math> is also called the ellipsoidal norm of Template:Math. For every ellipsoid, there are unique Template:Mvar and Template:Math that satisfy the above equation.<ref name=":0" />Template:Rp

The eigenvectors of Template:Mvar are the principal axes of the ellipsoid, and the eigenvalues of Template:Mvar are the reciprocals of the squares of the semi-axes (in three dimensions these are Template:Math, Template:Math and Template:Math).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }} pp. 17–18.</ref> In particular:

• The diameter of the ellipsoid is twice the longest semi-axis, which is twice the square-root of the reciprocal of the largest eigenvalue of Template:Mvar.
• The width of the ellipsoid is twice the shortest semi-axis, which is twice the square-root of the reciprocal of the smallest eigenvalue of Template:Mvar.

An invertible linear transformation applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable rotation, a consequence of the polar decomposition (also, see spectral theorem). If the linear transformation is represented by a symmetric 3 × 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 semi-axes are computed from the eigenvalues. The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.

For every positive definite matrix <math>\boldsymbol{A}</math>, there exists a unique positive definite matrix denoted Template:Math, such that <math>\boldsymbol{A} = \boldsymbol{A}^{1/ 2}\boldsymbol{A}^{1/ 2}; </math> this notation is motivated by the fact that this matrix can be seen as the "positive square root" of <math>\boldsymbol{A}.</math> The ellipsoid defined by <math>(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) = 1</math> can also be presented as<ref name=":0">Template:Cite Geometric Algorithms and Combinatorial Optimization</ref>Template:Rp

<math>A^{-1/2}\cdot S(\mathbf{0},1) + \mathbf{v}</math>

where S(0,1) is the unit sphere around the origin.

Parametric representation

File:Ellipsoid-affin.svg

ellipsoid as an affine image of the unit sphere

The key to a parametric representation of an ellipsoid in general position is the alternative definition:

An ellipsoid is an affine image of the unit sphere.

An affine transformation can be represented by a translation with a vector Template:Math and a regular 3 × 3 matrix Template:Math:

<math>\mathbf x \mapsto \mathbf f_0 + \boldsymbol A \mathbf x = \mathbf f_0 + x\mathbf f_1 + y\mathbf f_2 + z\mathbf f_3</math>

where Template:Math are the column vectors of matrix Template:Math.

A parametric representation of an ellipsoid in general position can be obtained by the parametric representation of a unit sphere (see above) and an affine transformation:

<math>\mathbf x(\theta, \varphi) = \mathbf f_0 + \mathbf f_1 \cos\theta \cos\varphi + \mathbf f_2 \cos\theta \sin\varphi + \mathbf f_3 \sin\theta, \qquad -\tfrac{\pi}{2} < \theta < \tfrac{\pi}{2},\quad 0 \le \varphi < 2\pi</math>.

If the vectors Template:Math form an orthogonal system, the six points with vectors Template:Math are the vertices of the ellipsoid and Template:Math are the semi-principal axes.

A surface normal vector at point Template:Math is

<math>\mathbf n(\theta, \varphi) = \mathbf f_2 \times \mathbf f_3\cos\theta\cos\varphi + \mathbf f_3 \times \mathbf f_1\cos\theta\sin\varphi + \mathbf f_1 \times \mathbf f_2\sin\theta.</math>

For any ellipsoid there exists an implicit representation Template:Math. If for simplicity the center of the ellipsoid is the origin, Template:Math, the following equation describes the ellipsoid above:<ref>Computerunterstützte Darstellende und Konstruktive Geometrie. Template:Webarchive Uni Darmstadt (PDF; 3,4 MB), S. 88.</ref>

<math>F(x, y, z) = \operatorname{det}\left(\mathbf x, \mathbf f_2, \mathbf f_3\right)^2 + \operatorname{det}\left(\mathbf f_1,\mathbf x, \mathbf f_3\right)^2 + \operatorname{det}\left(\mathbf f_1, \mathbf f_2, \mathbf x\right)^2 - \operatorname{det}\left(\mathbf f_1, \mathbf f_2, \mathbf f_3\right)^2 = 0</math>

Applications

The ellipsoidal shape finds many practical applications:

Geodesy

• Earth ellipsoid, a mathematical figure approximating the shape of the Earth.
• Reference ellipsoid, a mathematical figure approximating the shape of planetary bodies in general.

Mechanics

• Poinsot's ellipsoid, a geometrical method for visualizing the torque-free motion of a rotating rigid body.
• Lamé's stress ellipsoid, an alternative to Mohr's circle for the graphical representation of the stress state at a point.
• Manipulability ellipsoid, used to describe a robot's freedom of motion.
• Jacobi ellipsoid, a triaxial ellipsoid formed by a rotating fluid

Crystallography

• Index ellipsoid, a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal.
• Thermal ellipsoid, ellipsoids used in crystallography to indicate the magnitudes and directions of the thermal vibration of atoms in crystal structures.

Computer science

• Ellipsoid method, a convex optimization algorithm of theoretical significance

Lighting

• Ellipsoidal reflector floodlight
• Ellipsoidal reflector spotlight

Medicine

• Measurements obtained from MRI imaging of the prostate can be used to determine the volume of the gland using the approximation Template:Math (where 0.52 is an approximation for Template:Sfrac)<ref>Template:Cite journal</ref>

Dynamical properties

The mass of an ellipsoid of uniform density Template:Mvar is

<math>m = V \rho = \tfrac{4}{3} \pi abc \rho.</math>

The moments of inertia of an ellipsoid of uniform density are

<math>\begin{align}

 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.

\end{align}</math>

For Template:Math these moments of inertia reduce to those for a sphere of uniform density.

File:2003EL61art.jpg

Artist's conception of Template:Dp, a Jacobi-ellipsoid dwarf planet, with its two moons

Ellipsoids and cuboids rotate stably along their major or minor axes, but not along their median axis. This can be seen experimentally by throwing an eraser with some spin. In addition, moment of inertia considerations mean that rotation along the major axis is more easily perturbed than rotation along the minor axis.<ref>Goldstein, H G (1980). Classical Mechanics, (2nd edition) Chapter 5.</ref>

One practical effect of this is that scalene astronomical bodies such as Template:Dp generally rotate along their minor axes (as does Earth, which is merely oblate); in addition, because of tidal locking, moons in synchronous orbit such as Mimas orbit with their major axis aligned radially to their planet.

A spinning body of homogeneous self-gravitating fluid will assume the form of either a Maclaurin spheroid (oblate spheroid) or Jacobi ellipsoid (scalene ellipsoid) when in hydrostatic equilibrium, and for moderate rates of rotation. At faster rotations, non-ellipsoidal piriform or oviform shapes can be expected, but these are not stable.

Fluid dynamics

The ellipsoid is the most general shape for which it has been possible to calculate the creeping flow of fluid around the solid shape. The calculations include the force required to translate through a fluid and to rotate within it. Applications include determining the size and shape of large molecules, the sinking rate of small particles, and the swimming abilities of microorganisms.<ref>Dusenbery, David B. (2009).Living at Micro Scale, Harvard University Press, Cambridge, Massachusetts Template:Isbn.</ref>

In probability and statistics

The elliptical distributions, which generalize the multivariate normal distribution and are used in finance, can be defined in terms of their density functions. When they exist, the density functions Template:Mvar have the structure:

<math>f(x) = k \cdot g\left((\mathbf x - \boldsymbol\mu)\boldsymbol\Sigma^{-1}(\mathbf x - \boldsymbol\mu)^\mathsf{T}\right)</math>

where Template:Mvar is a scale factor, Template:Math is an Template:Mvar-dimensional random row vector with median vector Template:Math (which is also the mean vector if the latter exists), Template:Math is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and Template:Mvar is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.<ref>Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.</ref> The multivariate normal distribution is the special case in which Template:Math for quadratic form Template:Mvar.

Thus the density function is a scalar-to-scalar transformation of a quadric expression. Moreover, the equation for any iso-density surface states that the quadric expression equals some constant specific to that value of the density, and the iso-density surface is an ellipsoid.

See also

• Ellipsoidal dome
• Ellipsoidal coordinates
• Elliptical distribution, in statistics
• Flattening, also called ellipticity and oblateness, is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid), respectively.
• Focaloid, a shell bounded by two concentric, confocal ellipsoids
• Geodesics on an ellipsoid
• Geodetic datum, the gravitational Earth modeled by a best-fitted ellipsoid
• Homoeoid, a shell bounded by two concentric similar ellipsoids
• John ellipsoid, the smallest ellipsoid containing a given convex set.
• List of surfaces
• Superellipsoid

Notes

<references />

References

• Template:Citation

External links

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