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 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 a prolate spheroid. If the three axes have the same length, the ellipsoid is a sphere.
Examples of ellipsoids with equation sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, c=3), tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3)
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 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.
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 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, 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. Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see Circular section).
Determining the ellipse of a plane section
Plane section of an ellipsoid (See example)
Given: Ellipsoid and the plane with equation which have an ellipse in common. Wanted: Three vectors (center) and (conjugate vectors), such that the ellipse can be represented by the parametric equation
Solution: The scaling transforms the ellipsoid onto the unit sphere and the given plane onto the plane with equation . Let be the HESSE normal form of the new plane and its unit normal vector. Hence is the center of the intersection circle and its radius (See diagram).
In case of let be (The plane is horizontal !)
In case of let be
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
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
In general position
More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the solutions x to the equation
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.
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 is a scale factor, is an -dimensional random row vector with median vector (which is also the mean vector if the latter exists), is a positive definite matrix which is proportional to the covariance matrix if the latter exists, and 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 for quadratic form .
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.
^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 "Archived copy". Archived from the original on 2012-12-02. Retrieved 2012-01-08. (see next reference).
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