An ellipsoid is a closed quadric surface that is a three-dimensional analogue of an ellipse.
The standard equation of an ellipsoid centered at the origin of a Cartesian coordinate system
and aligned with the axes is
The points, and lie on the surface and the line segments from the origin to these points
are called the semi-principal axes of length a, b, c. They correspond to the semi-major
axis and semi-minor axis of the appropriate ellipses.
There are four distinct cases of which one is degenerate:
— tri-axial or scalene ellipsoid; — oblate ellipsoid of revolution;
— prolate ellipsoid of revolution; — the degenerate case of a sphere;
Mathematical literature often uses 'ellipsoid' in place of 'tri-axial ellipsoid'. Scientific
literature often uses 'ellipsoid' in place of 'ellipsoid of revolution' and only applies
the adjective 'tri-axial' when treating the general case. Older literature uses 'spheroid'
in place of 'ellipsoid of revolution'. Any planar cross section passing through the
center of an ellipsoid forms an ellipse on its surface: this degenerates to a circle
for sections normal to the symmetry axis of an ellipsoid of revolution
Generalised equations More generally, an arbitrarily oriented ellipsoid,
centered at v, is defined by the solutions x to the equation
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: , and . 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.
If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors
of the matrix are orthogonal 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.
Parameterization The surface of the ellipsoid may be parameterized
in several ways. One possible choice which singles out the 'z'-axis is:
where
The parameters may be interpreted as spherical coordinates. For constant u, that is on the
ellipse which is the intercept with a constant z plane, v then plays the role of the eccentric
anomaly for that ellipse. For constant v on a plane through the Oz axis the parameter
u plays the same role for the ellipse of intersection. Two other similar parameterizations are possible,
each with their own interpretations. Only on an ellipse of revolution can a unique definition
of reduced latitude be made. Volume and surface area
Volume The volume of the internal part of the ellipsoid
is
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 two thirds the volume of a circumscribed elliptic cylinder.
The volumes of the maximum inscribed and minimum circumscribed boxes are respectively:
The volume of an ellipse of dimension higher than 3 can be calculated using the dimensional
constant given for the volume of a hypersphere. One can also define ellipsoids in higher dimensions,
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.
Surface area The surface area of a general ellipsoid is
where
and where F(φ,k) and E(φ,k) are incomplete elliptic integrals of the first and second
kind respectively.[1] The surface area of an ellipsoid of revolution
may be expressed in terms of elementary functions:
which, as follows from basic trigonometric identities, are equivalent expressions. In
both cases e may again be identified as the eccentricity of the ellipse formed by the
cross section through the symmetry axis.. Derivations of these results may be found
in standard sources, for example Mathworld. Approximate formula
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.
Dynamical properties The mass of an ellipsoid of uniform density
ρ is:
The moments of inertia of an ellipsoid of uniform density are:
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; in addition,
because of tidal locking, moons in synchronous orbit such as Mimas orbit with their major
axis aligned radially to their planet. A relaxed ellipsoid, that is, one in hydrostatic
equilibrium, has an oblateness a − c directly proportional to its mean density and mean
radius. Ellipsoids with a differentiated interior—that is, a denser core than mantle—have a lower
oblateness than a homogeneous body. Over all, the ratio/(a−c) is approximately 0.25, though
this drops for rapidly rotating bodies. The terminology typically used for bodies
rotating on their minor axis and whose shape is determined by their gravitational field
is Maclaurin spheroid and Jacobi ellipsoid. At faster rotations, piriform or oviform shapes
can be expected, but these are not stable. Fluid properties
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. Equations in specific coordinate systems
Cartesian
Spherical
Cylindrical
See also Paraboloid
Poinsot's ellipsoid Hyperboloid
Reference ellipsoid Geoid
Ellipsoid method Superellipsoid
Haumea, a scalene-ellipsoid-shaped dwarf planet Homoeoid, a shell bounded by two concentric,
similar ellipsoids Focaloid, a shell bounded by two concentric,
confocal ellipsoids Elliptical distribution, in statistics
Ellipse References
^ http:see.stanford.edulsoeldsee263/15-symm.pdf ^ F. W. J. Olver, D. W. Lozier, R. F. Boisvert,
and C. W. Clark, editors, 2010, NIST Handbook of Mathematical Functions, available on line
at http:dlmf.nist.gov/19.33. ^ NIST athttp:www.nist.gov
^ Prolate Spheroid at Mathworld ^ Final answers by Gerard P. Michon. See Thomsen's
formulas and Cantrell's comments. ^ Goldstein, H G. Classical Mechanics, Chapter
5. ^ "Shapes of the Saturnian Icy Satellites".
^ Dusenbery, David B..Living at Micro Scale, Harvard University Press, Cambridge, Mass.
ISBN 978-0-674-03116-6. "Ellipsoid" by Jeff Bryant, Wolfram Demonstrations
Project, 2007. Ellipsoid and Quadratic Surface, MathWorld.
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