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 perpendicularaxes 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. If the third axis is shorter, the ellipsoid is an oblate spheroid; if it is longer, it is a prolate spheroid. If the three axes have the same length, the ellipsoid is a sphere.
The general ellipsoid, also known as triaxial ellipsoid, is a quadratic surface which is defined in Cartesian coordinates as:
where , and are the length of the semi-axes.
The points , and lie on the surface. The line segments from the origin to these points are called the principal semi-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.
and where F(φ, k) and E(φ, k) are incomplete elliptic integrals of the first and second kind respectively. The surface area of this general ellipsoid can also be expressed using the RF and RDCarlson symmetric forms of the elliptic integrals by simply substituting the above formula to the respective definitions:
Unlike the expression with F(φ, k) and E(φ, k), the variant based on the Carlson symmetric integrals yields valid results for a sphere and only the axis c must be the smallest, the order between the two larger axes, a and b can be arbitrary.
The surface area of an ellipsoid of revolution (or spheroid) may be expressed in terms of elementary functions:
which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for Soblate 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.
Here p ≈ 1.6075 yields a relative error of at most 1.061%; 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 and b, the area is approximately 2πab, equivalent to p = log23 ≈ 1.5849625007.
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. Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see Circular section).
Determining the ellipse of a plane sectionedit
Given: Ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 and the plane with equation nxx + nyy + nzz = d, which have an ellipse in common.
Wanted: Three vectors f0 (center) and f1, f2 (conjugate vectors), such that the ellipse can be represented by the parametric equation
In any case, the vectors e1, e2 are orthogonal, parallel to the intersection plane and have length ρ (radius of the circle). Hence the intersection circle can be described by the parametric equation
The reverse scaling (see above) transforms the unit sphere back to the ellipsoid and the vectors e0, e1, e2 are mapped onto vectors f0, f1, f2, 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 a = 4, b = 5, c = 3 which is cut by the plane x + y + z = 5.
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). Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898. 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, too.
Steps of the constructionedit
Choose an ellipseE and a hyperbolaH, which are a pair of focal conics:
with the vertices and foci of the ellipse
and a string (in diagram red) of length l.
Pin one end of the string to vertexS1 and the other to focus F2. The string is kept tight at a point P with positive y- and z-coordinates, such that the string runs from S1 to P behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from P to F2 runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance |S1P| 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.
Then: P is a point of the ellipsoid with equation
The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.
Equations for the semi-axes of the generated ellipsoid can be derived by special choices for point P:
The lower part of the diagram shows that F1 and F2 are the foci of the ellipse in the xy-plane, too. Hence, it is confocal to the given ellipse and the length of the string is l = 2rx + (a − c). Solving for rx yields rx = 1/2(l − a + c); furthermore r2 y = r2 x − c2.
From the upper diagram we see that S1 and S2 are the foci of the ellipse section of the ellipsoid in the xz-plane and that r2 z = r2 x − a2.
If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters a, b, l for a pins-and-string construction.
If E is an ellipsoid confocal to E with the squares of its semi-axes
then from the equations of E
one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes a, b, c as ellipsoid E. 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.
The converse statement is true, too: if one chooses a second string of length l and defines
then the equations
are valid, which means the two ellipsoids are confocal.
Limit case, ellipsoid of revolutionedit
In case of a = c (a spheroid) one gets S1 = F1 and S2 = F2, which means that 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 rotationally symmetric around the x-axis and
Properties of the focal hyperbolaedit
If one views an ellipsoid from an external point V of its focal hyperbola, than it seems to be a sphere, that is its apparent shape is a circle. Equivalently, the tangents of the ellipsoid containing point V are the lines of a circular cone, whose axis of rotation is the tangent line of the hyperbola at V. If one allows the center V to disappear into infinity, one gets an orthogonalparallel 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 V and main point H on the tangent of the hyperbola at point V. (H is the foot of the perpendicular from V onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin O is the circle's center; in the central case main point H is the center.
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 a, b for rz → 0. For the limit case one gets
In general positionedit
As a quadricedit
If v is a point and A is a real, symmetric, positive-definite matrix, then the set of points x that satisfy the equation
is an ellipsoid centered at v. The eigenvectors of A are 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.
For a = b = c these moments of inertia reduce to those for a sphere of uniform density.
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.
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.
where k is a scale factor, x is an n-dimensional random row vector with median vector μ (which is also the mean vector if the latter exists), Σ 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. The multivariate normal distribution is the special case in which g(z) = exp(−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.
^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 online at "DLMF: 19.33 Triaxial Ellipsoids". Archived from the original on 2012-12-02. Retrieved 2012-01-08. (see next reference).
^NIST (National Institute of Standards and Technology) at http://www.nist.gov Archived 2015-06-17 at the Wayback Machine