Ellipsoid

examples of ellipsoids with equation

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

sphere (top, a=4),
spheroid (bottom left, a=b=5, c=3),
tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3)

An ellipsoid is a surface that may 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 ellipsoid is said to be tri-axial or rarely scalene, 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. If the third axis is shorter, the ellipsoid is an oblate spheroid, if it is longer, it is prolate spheroid. If the three axes have the same length, the ellipsoid is a sphere.

Standard equation

When choosing a Cartesian coordinate system, such that the origin is the center of the ellipsoid, and the coordinate axes are axes of the ellipsoid, the implicit equation of the ellipsoid has the standard form

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

where $a, b, c$ are positive real numbers.

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 semi-principal axes of the ellipsoid, because $a, b, c$ are half the length of the principal axes. They correspond to the semi-major axis and semi-minor axis of an ellipse.

If $a=b>c,$ one has an oblate spheroid; if $a=b<c,$ one has a prolate spheroid; if $a=b=c,$ one has a sphere.

Plane sections

plane section of an ellipsoid

The intersection of a plane and a sphere is a circle if it is neither empty nor reduced to a single point. Because any ellipsoid is an affine image of the unit sphere and circles are mapped onto ellipses by affine transformations, the intersection of a plane with an ellipsoid is an ellipse or a simple point, or is empty. Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see circular section).

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

{\begin{aligned}x&=a\cos(\theta )\cos(\varphi ),\\y&=b\cos(\theta )\sin(\varphi ),\\z&=c\sin(\theta ),\end{aligned}}\,\!

where

-{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}},\qquad -\pi \leq \varphi \leq \pi .

These parameters may be interpreted as spherical coordinates, where $\pi /2-\theta$ is the polar angle, and $\varphi$ is the azimuth angle of the point $(x, y, z)$ of the ellipsoid.

In general position

As quadric

More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the solutions x to the equation

(\mathbf {x-v} )^{\mathrm {T} }\!A\,(\mathbf {x-v} )=1,

where A is a positive definite matrix and x, v are vectors.

The eigenvectors of A define the principal axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes: $a^{{-2}}$ , $b^{-2}$ and $c^{{-2}}$ .^[1] 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-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 singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.

Parametric representation

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 ${\vec {f}}_{0}$ and a regular 3×3-matrix $A$ :

{\vec {x}}\mapsto {\vec {f}}_{0}+A{\vec {x}}={\vec {f}}_{0}+x{\vec {f}}_{1}+y{\vec {f}}_{2}+z{\vec {f}}_{3}

where ${\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}$ are the column vectors of matrix $A$ .

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:

${\vec {x}}(\theta ,\varphi )={\vec {f}}_{0}+{\vec {f}}_{1}\cos \theta \cos \varphi +{\vec {f}}_{2}\cos \theta \sin \varphi +{\vec {f}}_{3}\sin \theta ,\quad -\pi /2\leq \theta \leq \pi /2,\ 0\leq \varphi <2\pi$ .

If the vectors ${\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3}$ form an orthogonal system, the points with vectors ${\vec {f}}_{0}\pm {\vec {f}}_{i},\ i=1,2,3,$ are the vertices of the ellipsoid and $|{\vec {f}}_{1}|,|{\vec {f}}_{2}|,|{\vec {f}}_{3}|$ are the semi principal axes.

A surface normal vector at point ${\vec {x}}(\theta ,\varphi )$ is

{\vec {n}}(\theta ,\varphi )={\vec {f}}_{2}\times {\vec {f}}_{3}\cos \theta \cos \varphi +{\vec {f}}_{3}\times {\vec {f}}_{1}\cos \theta \sin \varphi +{\vec {f}}_{1}\times {\vec {f}}_{2}\sin \theta .

For any ellipsoid there exists an implicit representation $F(x,y,z)=0$ . If for simplicity the center of the ellipsoid is the origin, i.e. ${\vec {f}}_{0}=(0,0,0)^{T}$ , the following equation describes the ellipsoid above:^[2]

F(x,y,z)=\operatorname {det} ({\vec {x}},{\vec {f}}_{2},{\vec {f}}_{3})^{2}\;+\;\operatorname {det} ({\vec {f}}_{1},{\vec {x}},{\vec {f}}_{3})^{2}\;+\;\operatorname {det} ({\vec {f}}_{1},{\vec {f}}_{2},{\vec {x}})^{2}\;-\;\operatorname {det} ({\vec {f}}_{1},{\vec {f}}_{2},{\vec {f}}_{3})^{2}=0

Volume and surface area

Volume

The volume bounded by the ellipsoid is

V={\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.

The volume of an ellipsoid is ${\frac {2}{3}}$ the volume of a circumscribed elliptic cylinder, and $\frac{\pi}{6}$ 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.

Surface area

The surface area of a general (tri-axial) ellipsoid is^[3]^[4]

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),

where

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

and where F(φ,k) and E(φ,k) are incomplete elliptic integrals of the first and second kind respectively.

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 {(1-e^{2})}{e}}\cdot {\mbox{artanh}}(e)\right)\quad {\mbox{where}}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}\quad (c<a),

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

which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for $S_{\rm {oblate}}$ can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases e 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.^[5]

Approximate formula

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

Here p ≈ 1.6075 yields a relative error of at most 1.061%;^[6] a value of p = 8/5 = 1.6 is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.

In the "flat" limit of c much smaller than a, b, the area is approximately 2 $π$ ab.

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 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.

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.

Lighting

Dynamical properties

The mass of an ellipsoid of uniform density ρ is:

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

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

I_{\mathrm {xx} }={\frac {m(b^{2}+c^{2})}{5}},\qquad I_{\mathrm {yy} }={\frac {m(c^{2}+a^{2})}{5}},\qquad I_{\mathrm {zz} }={\frac {m(a^{2}+b^{2})}{5}},

I_{\mathrm {xy} }=I_{\mathrm {yz} }=I_{\mathrm {zx} }=0.\,\!

For $a=b=c$ these moments of inertia reduce to those for a sphere of uniform density.

Artist's conception of Haumea, 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.^[7]

One practical effect of this is that scalene astronomical bodies such as Haumea 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.^[8]

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 f have the structure:

f(x)=k\cdot g((x-\mu )\Sigma ^{-1}(x-\mu )^{\mathrm {T} })

where $k$ is a scale factor, $x$ is an $n$ -dimensional random row vector with median vector $\mu$ (which is also the mean vector if the latter exists), $\Sigma$ is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and $g$ is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.^[9] The multivariate normal distribution is the special case in which $g(z)=e^{-z/2}$ for quadratic form $z$ .

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.

In higher dimensions

The volume of a higher-dimensional ellipsoid (a hyperellipsoid) can be calculated using the dimensional constant given for the volume of a hypersphere. One can also define hyperellipsoids 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.

References

↑ http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf
↑ Computerunterstützte Darstellende und Konstruktive Geometrie. Uni Darmstadt (PDF; 3,4 MB), S. 88.
↑ F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors, 2010, NIST Handbook of Mathematical Functions (Cambridge University Press), available on line at http://dlmf.nist.gov/19.33 (see next reference).
↑ NIST (National Institute of Standards and Technology) at http://www.nist.gov
↑ Prolate Spheroid at Mathworld
↑ Final answers by Gerard P. Michon (2004-05-13). See Thomsen's formulas and Cantrell's comments.
↑ Goldstein, H G (1980). Classical Mechanics, (2nd edition) Chapter 5.
↑ Dusenbery, David B. (2009).Living at Micro Scale, Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.
↑ Frahm, G., Junker, M., & Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statistics & Probability Letters, 63(3), 275–286.

External links

Wikimedia Commons has media related to Ellipsoids.

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

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