Ellipsoid

An ellipsoid is a surface that can 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. 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, 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. In the case of two axes being the same length:

If the third axis is shorter, the ellipsoid is a sphere that has been flattened.
If the third axis is longer, it is a sphere that has been lengthened.

If the three axes have the same length, the ellipsoid is a sphere.

Standard equation

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 are half the length of the principal axes. They correspond to the semi-major axis and semi-minor axis of an ellipse.
In spherical coordinate system for which, the general ellipsoid is defined as:
where is the polar angle and is the azimuthal angle.
When, the ellipsoid is a sphere.
When, the ellipsoid is a spheroid or ellipsoid of revolution. In particular, if, it is an oblate spheroid; if, it is a prolate spheroid.

Parameterization

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
where
These parameters may be interpreted as spherical coordinates, where is the polar angle and is the azimuth angle of the point of the ellipsoid.
Measuring from the equator rather than a pole,
where
is the reduced latitude, parametric latitude, or eccentric anomaly and is azimuth or longitude.
Measuring angles directly to the surface of the ellipsoid, not to the circumscribed sphere,
where
would be geocentric latitude on the Earth, and is longitude. These are true spherical coordinates with the origin at the center of the ellipsoid.
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.

Volume

The volume bounded by the ellipsoid is
In terms of the principal diameters , 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 the volume of a circumscribed elliptic cylinder, and the volume of the circumscribed box. The volumes of the inscribed and circumscribed boxes are respectively:

Surface area

The surface area of a general ellipsoid is
where
and where and are incomplete elliptic integrals of the first and second kind respectively.
The surface area of this general ellipsoid can also be expressed in terms of, one of the Carlson symmetric forms of elliptic integrals:
Simplifying the above formula using properties of, this can also be expressed in terms of the volume of the ellipsoid :
Unlike the expression with and, the equations in terms of do not depend on the choice of an order on,, and.
The surface area of an ellipsoid of revolution may be expressed in terms of elementary functions:
or
or
and
which, as follows from basic trigonometric identities, are equivalent expressions. In both cases 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 yields a relative error of at most 1.061%; a value of is optimal for nearly spherical ellipsoids, with a relative error of at most 1.178%.
In the "flat" limit of much smaller than and, the area is approximately, equivalent to.

Plane sections

The intersection of a plane and a sphere is a circle. 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.

Determining the ellipse of a plane section

Given: Ellipsoid and the plane with equation, which have an ellipse in common.
Wanted: Three vectors and, , 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.
Where , let
Where, let
In any case, the vectors are orthogonal, parallel to the intersection plane and have length . Hence the intersection circle can be described by the parametric equation
The reverse scaling 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.

Pins-and-string construction

The pins-and-string construction of an ellipsoid is a transfer of the idea constructing an ellipse using two pins and a string.
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. Main investigations and the extension to quadrics was done by the German mathematician O. Staude in 1882, 1886 and 1898. A description of the pins-and-string construction of ellipsoids and hyperboloids is contained in the book Geometry and the Imagination by Hilbert & Cohn-Vossen.

Steps of the construction

Choose an ellipse and a hyperbola, which are a pair of focal conics: with the vertices and foci of the ellipse and a string of length.
Pin one end of the string to vertex and the other to focus. The string is kept tight at a point with positive - and -coordinates, such that the string runs from to behind the upper part of the hyperbola and is free to slide on the hyperbola. The part of the string from to runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance 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: 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.
Semi-axes

Equations for the semi-axes of the generated ellipsoid can be derived by special choices for point :
The lower part of the diagram shows that and are the foci of the ellipse in the -plane, too. Hence, it is confocal to the given ellipse and the length of the string is. Solving for yields ; furthermore.
From the upper diagram we see that and are the foci of the ellipse section of the ellipsoid in the -plane and that.

Converse

If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters,, for a pins-and-string construction.

Confocal ellipsoids

If is an ellipsoid confocal to with the squares of its semi-axes
then from the equations of
one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes as ellipsoid. Therefore one considers the focal conics of a triaxial ellipsoid as the foci and calls them the focal curves of the ellipsoid.
The converse statement is true, too: if one chooses a second string of length and defines
then the equations
are valid, which means the two ellipsoids are confocal.

Limit case, ellipsoid of revolution

In case of one gets and, which means that the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on the -axis. The ellipsoid is rotationally symmetric around the -axis and

Properties of the focal hyperbola

; True curve
; Umbilical points

Property of the focal ellipse

The focal ellipse together with its inner part can be considered as the limit surface of the pencil of confocal ellipsoids determined by for. For the limit case one gets