Ellipsoid

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 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.

Using a Cartesian coordinate system in which 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

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.

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.

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

These parameters may be interpreted as spherical coordinates, where θ is the polar angle and φ is the azimuth angle of the point (x, y, z) of the ellipsoid.[1]

θ is the reduced latitude, parametric latitude, or eccentric anomaly and λ is azimuth or longitude.

Measuring angles directly to the surface of the ellipsoid, not to the circumscribed sphere,

γ would be geocentric latitude on the Earth, and λ is longitude. These are true spherical coordinates with the origin at the center of the ellipsoid.[citation needed]

In geodesy, the geodetic latitude is most commonly used, as the angle between the vertical and the equatorial plane, defined for a biaxial ellipsoid. For a more general triaxial ellipsoid, see ellipsoidal latitude.

In terms of the principal diameters A, B, C (where A = 2a, B = 2b, C = 2c), the volume is

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 2/3 the volume of a circumscribed elliptic cylinder, and π/6 the volume of the circumscribed box. The volumes of the inscribed and circumscribed boxes are respectively:

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:

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.[5]

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

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

Solution: The scaling u = x/a, v = y/b, w = z/c transforms the ellipsoid onto the unit sphere u2 + v2 + w2 = 1 and the given plane onto the plane with 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).[8] Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898.[9][10][11] 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,[12] too.

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 + (ac). Solving for rx yields rx = 1/2(la + 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.

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.[13]

The converse statement is true, too: if one chooses a second string of length l and defines

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

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

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.[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 × 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.

The key to a parametric representation of an ellipsoid in general position is the alternative definition:

An affine transformation can be represented by a translation with a vector f0 and a regular 3 × 3 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:

If the vectors f1, f2, f3 form an orthogonal system, the six points with vectors f0 ± f1,2,3 are the vertices of the ellipsoid and |f1|, |f2|, |f3| are the semi-principal axes.

For any ellipsoid there exists an implicit representation F(x, y, z) = 0. If for simplicity the center of the ellipsoid is the origin, f0 = 0, the following equation describes the ellipsoid above:[18]

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.[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:

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.[22] 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.

One can also define a hyperellipsoid as the image of a sphere under an invertible affine transformation. The spectral theorem can again be used to obtain a standard equation of the form

The volume of an n-dimensional hyperellipsoid can be obtained by replacing Rn by the product of the semi-axes a1a2...an in the formula for the volume of a hypersphere: