Ellipsoid

The volume bounded by the ellipsoid is

${\displaystyle V={\frac {4}{3}}\pi abc.}$

Alternatively expressed, where A, B and C are the lengths of the principal axes (A = 2a, B = 2b and C = 2c):

${\displaystyle V={\frac {\pi }{6}}ABC}$.

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 ${\textstyle {\frac {2}{3}}}$ the volume of a circumscribed elliptic cylinder, and ${\textstyle {\frac {\pi }{6}}}$ the volume of the circumscribed box.

The volumes of the inscribed and circumscribed boxes are respectively:

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

The surface area of a general (tri-axial) ellipsoid is[2][3]

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

where

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

and where F(φ, k) and E(φ, k) are incomplete elliptic integrals of the first and second kind respectively.[4]

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

${\displaystyle {\begin{aligned}S_{\text{oblate}}&=2\pi a^{2}\left(1+{\frac {c^{2}}{ea^{2}}}\cdot \operatorname {arctanh} (e)\right)&&{\text{where }}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}\quad (c<a),\\S_{\text{prolate}}&=2\pi a^{2}\left(1+{\frac {c}{ae}}\cdot \arcsin(e)\right)&&{\text{where }}\quad e^{2}=1-{\frac {a^{2}}{c^{2}}}\quad (c>a),\end{aligned}}}$

which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for ${\displaystyle S_{\text{oblate}}}$ 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]

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

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 2πab, equivalent to p ≈ 1.5850.

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.[7] Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see Circular section).

Plane section of an ellipsoid (see example)

Given: Ellipsoid ${\textstyle \ {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1\ }$ and the plane with equation ${\textstyle \ n_{x}x+n_{y}y+n_{z}z=d\,,}$ which have an ellipse in common.

Wanted: Three vectors ${\displaystyle {\vec {f}}_{0}}$ (center) and ${\displaystyle {\vec {f}}_{1},\;{\vec {f}}_{2}}$ (conjugate vectors), such that the ellipse can be represented by the parametric equation

${\displaystyle {\vec {x}}={\vec {f}}_{0}+{\vec {f}}_{1}\cos t+{\vec {f}}_{2}\sin t\quad }$ (see ellipse).

Plane section of the unit sphere (see example)

Solution: The scaling ${\textstyle \ u={\frac {x}{a}}\,,\ v={\frac {y}{b}}\,,\ w={\frac {z}{c}}\ }$ transforms the ellipsoid onto the unit sphere ${\displaystyle u^{2}+v^{2}+w^{2}=1\ }$ and the given plane onto the plane with equation ${\displaystyle \ n_{x}au+n_{y}bv+n_{z}cw=d\ }$. Let ${\displaystyle \ m_{u}u+m_{v}v+m_{w}w=\delta \ }$ be the Hesse normal form of the new plane and ${\displaystyle \;{\vec {m}}={\begin{bmatrix}m_{u}&m_{v}&m_{w}\end{bmatrix}}^{\mathsf {T}}\;}$ its unit normal vector. Hence ${\displaystyle {\vec {e}}_{0}=\delta \;{\vec {m}}\;}$ is the center of the intersection circle and ${\displaystyle \;\rho ={\sqrt {1-\delta ^{2}}}\;}$ its radius (see diagram).

Where ${\displaystyle \ m_{w}=\pm 1\,}$, let ${\displaystyle \ {\vec {e}}_{1}={\begin{bmatrix}\rho &0&0\end{bmatrix}}^{\mathsf {T}},\ {\vec {e}}_{2}={\begin{bmatrix}0&\rho &0\end{bmatrix}}^{\mathsf {T}}.}$ (The plane is horizontal!)

Where ${\displaystyle \ m_{w}\neq \pm 1\,}$, let ${\textstyle \ {\vec {e}}_{1}=\rho \,{\frac {{\begin{bmatrix}m_{v}&-m_{u}&0\end{bmatrix}}^{\mathsf {T}}}{\sqrt {m_{u}^{2}+m_{v}^{2}}}}\,,\ {\vec {e}}_{2}={\vec {m}}\times {\vec {e}}_{1}\ .}$

In any case, the vectors ${\displaystyle {\vec {e}}_{1},{\vec {e}}_{2}}$ are orthogonal, parallel to the intersection plane and have length ${\displaystyle \rho }$ (radius of the circle). Hence the intersection circle can be described by the parametric equation ${\displaystyle \;{\vec {u}}={\vec {e}}_{0}+{\vec {e}}_{1}\cos t+{\vec {e}}_{2}\sin t\;.}$

The reverse scaling (see above) transforms the unit sphere back to the ellipsoid and the vectors ${\displaystyle {\vec {e}}_{0},{\vec {e}}_{1},{\vec {e}}_{2}}$ are mapped onto vectors ${\displaystyle {\vec {f}}_{0},{\vec {f}}_{1},{\vec {f}}_{2}}$, 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 ${\displaystyle \;a=4,\;b=5,\;c=3\;}$ which is cut by the plane ${\displaystyle \;x+y+z=5\;.}$

Equations for the semi-axes of the generated ellipsoid can be derived by special choices for point ${\displaystyle P}$: ${\displaystyle Y=(0,r_{y},0),\ Z=(0,0,r_{z})}$.

The lower part of the diagram shows: ${\displaystyle F_{1},F_{2}}$ are the foci of the ellipse in the x-y-plane, too. Hence, it is confocal to the given ellipse and the length of the string is ${\displaystyle l=2r_{x}+(a-c)}$. Solving for ${\displaystyle r_{x}}$ yields: ${\textstyle r_{x}={\frac {1}{2}}(l-a+c)}$. Further more: ${\displaystyle r_{y}^{2}=r_{x}^{2}-c^{2}}$.

From the upper diagram one gets: ${\displaystyle S_{1},S_{2}}$ are the foci of the ellipse (of the ellipsoid) in the x-z-plane and the equation ${\displaystyle r_{z}^{2}=r_{x}^{2}-a^{2}}$.

If, conversely, a 3-axial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters ${\displaystyle a,b,l}$ for a pins-and-string construction.

If ${\displaystyle {\mathcal {\overline {E}}}}$ is an ellipsoid confocal to ${\displaystyle {\mathcal {E}}}$ with the squares of its semi-axes

${\displaystyle {\begin{aligned}{\overline {r}}_{x}^{2}&=r_{x}^{2}-\lambda ,&{\overline {r}}_{y}^{2}&=r_{y}^{2}-\lambda ,&{\overline {r}}_{z}^{2}&=r_{z}^{2}-\lambda \;,\end{aligned}}}$

then from the equations of ${\displaystyle {\mathcal {E}}}$

${\displaystyle {\begin{aligned}r_{x}^{2}-r_{y}^{2}&=c^{2},&r_{x}^{2}-r_{z}^{2}&=a^{2},&r_{y}^{2}-r_{z}^{2}&=a^{2}-c^{2}=b^{2},\end{aligned}}}$

one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes ${\displaystyle a,b,c}$ as ellipsoid ${\displaystyle {\mathcal {E}}}$. Therefore (analogously to the foci of an ellipse) one considers the focal conics of a 3-axial ellipsoid as the (infinite many) foci and calls them the focal curves of the ellipsoid.[13]

The converse statement is true, too: if one chooses a second string of length ${\displaystyle {\overline {l}}}$ and defines ${\displaystyle \lambda =r_{x}^{2}-{\overline {r}}_{x}^{2}}$ then the equations ${\displaystyle {\overline {r}}_{y}^{2}=r_{y}^{2}-\lambda ,\ {\overline {r}}_{z}^{2}=r_{z}^{2}-\lambda }$ are valid, which means the two ellipsoids are confocal.

In case of ${\displaystyle a=c}$ one gets ${\displaystyle S_{1}=F_{1},\;S_{2}=F_{2}}$, which means: the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on the x-axis. The ellipsoid is rotational symmetric with the x-axis as axis of rotation and ${\textstyle r_{x}={\frac {l}{2}},\ r_{y}=r_{z}={\sqrt {r_{x}^{2}-c^{2}}}}$.

Top: 3-axial Ellipsoid with its focal hyperbola.
Below: parallel-/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 ${\displaystyle V}$ of its focal hyperbola, than it seems to be a sphere, i.e. the apparent shape is a circle. Or equivalent: The tangents of the ellipsoid containing point ${\displaystyle V}$ are the lines of a circular cone, whose axis of rotation is the tangent of the hyperbola at ${\displaystyle V}$.[14][15] If one allows the center ${\displaystyle V}$ to disappear into infinity, one gets an orthogonal parallel projection with the corresponding asymptote of the focal hyperbola as direction. The true curve of shape (tangent points) on the ellipsoid is i.g. no circle!

The lower part of the diagram shows on the left a parallel projection of an ellipsoid (semi-axes: 60, 40, 30) along an asymptote and on the right a central projection with center ${\displaystyle V}$ and main point ${\displaystyle H}$ on the tangent of the hyperbola at point ${\displaystyle V}$. (${\displaystyle H}$ is the foot of the perpendicular from ${\displaystyle V}$ onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin ${\displaystyle O}$ is the circle's center, in the central case main point ${\displaystyle H}$ is the center.

Umbical points

The focal hyperbola intersects the ellipsoid at its 4 umbilical points.[16]

The focal ellipse together with its inner part can be considered as the limit surface (infinitely thin ellipsoid) of the pencil of confocal ellipsoids determined by ${\displaystyle a,b}$ for ${\displaystyle r_{z}\to 0}$. For the limit case one gets ${\displaystyle r_{x}=a,\;r_{y}=b,\;l=3a-c.}$

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

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

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: ${\displaystyle a^{-2}}$, ${\displaystyle b^{-2}}$ and ${\displaystyle c^{-2}}$.[17] 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 semi-axes are computed from the eigenvalues. The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.

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

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

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

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:

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

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

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

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

For any ellipsoid there exists an implicit representation ${\displaystyle F(x,y,z)=0}$. If for simplicity the center of the ellipsoid is the origin, i.e. ${\displaystyle {\vec {f}}_{0}={\begin{bmatrix}0&0&0\end{bmatrix}}^{\mathsf {T}}}$, the following equation describes the ellipsoid above:[18]

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

The mass of an ellipsoid of uniform density ρ is:

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

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

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

For ${\displaystyle a=b=c}$ 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.[20]

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.

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.[21]

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:

${\displaystyle f(x)=k\cdot g\left((x-\mu )\Sigma ^{-1}(x-\mu )^{\mathsf {T}}\right)}$

where ${\displaystyle k}$ is a scale factor, ${\displaystyle x}$ is an ${\displaystyle n}$-dimensional random row vector with median vector ${\displaystyle \mu }$ (which is also the mean vector if the latter exists), ${\displaystyle \Sigma }$ is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and ${\displaystyle g}$ is a function mapping from the non-negative reals to the non-negative reals giving a finite area under the curve.[22] The multivariate normal distribution is the special case in which ${\displaystyle g(z)=e^{-z/2}}$ for quadratic form ${\displaystyle z}$.

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.