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Ellipsoid

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ellipsoid

The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by

(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1,
(1)

where the semi-axes are of lengths a, b, and c. In spherical coordinates, this becomes

(r^2cos^2thetasin^2phi)/(a^2)+(r^2sin^2thetasin^2phi)/(b^2)+(r^2cos^2phi)/(c^2)=1.
(2)

Tietze (1965, p. 28) calls the general ellipsoid with a!=b!=c a "triaxial ellipsoid." If the lengths of two axes of an ellipsoid are the same, the figure is called an ellipsoid of revolution or spheroid. Denote the equal semi-axes lengths of a spheroid a=b, call a the equatorial radius, and call the other semi-axis length the polar radius c. Then if a>c, the spheroid is called an oblate spheroid, and if a<c, the spheroid is called an prolate spheroid. If all three semi-axes lengths are the same so a=b=c, the ellipsoid is a sphere.

There are two families of parallel circular cross sections in every ellipsoid. However, the two coincide for spheroids (Hilbert and Cohn-Vossen 1999, pp. 17-19). If the two sets of circles are fastened together by suitably chosen slits so that they are free to rotate without sliding, the model is movable. Furthermore, the disks can always be moved into the shape of a sphere (Hilbert and Cohn-Vossen 1999, p. 18).

In 1882, Staude discovered a "thread" construction for an ellipsoid analogous to the taut pencil and string construction of the ellipse (Hilbert and Cohn-Vossen 1999, pp. 19-22). This construction makes use of a fixed framework consisting of an ellipse and a hyperbola.

The parametric equations of an ellipsoid can be written as

x=acosusinv
(3)
y=bsinusinv
(4)
z=ccosv.
(5)

for u in [0,2pi) and v in [0,pi].

In this parametrization, the coefficients of the first fundamental form are

E=(b^2cos^2u+a^2sin^2u)sin^2v
(6)
F=(b^2-a^2)cosusinucosvsinv
(7)
G=(a^2cos^2u+b^2sin^2u)cos^2v+c^2sin^2v,
(8)

and of the second fundamental form are

e=(abcsin^2v)/(sqrt(a^2b^2cos^2v+c^2(b^2cos^2u+a^2sin^2u)sin^2v))
(9)
f=0
(10)
g=(abc)/(sqrt(a^2b^2cos^2v+c^2(b^2cos^2u+a^2sin^2u)sin^2v)).
(11)

Also in this parametrization, the Gaussian curvature is

K=(a^2b^2c^2)/([a^2b^2cos^2v+c^2(b^2cos^2u+a^2sin^2u)sin^2v]^2)
(12)

and the mean curvature is

H=(abc[3(a^2+b^2)+2c^2+(a^2+b^2-2c^2)cos(2v)-2(a^2-b^2)cos(2u)sin^2v])/(8[a^2b^2cos^2v+c^2(b^2cos^2u+a^2sin^2u)sin^2v]^(3/2)).
(13)

The Gaussian curvature can be given implicitly by

K(x,y,z)=(a^2b^6c^6)/([c^4b^4+c^4(a^2-b^2)y^2+b^4(a^2-c^2)z^2]^2)
(14)
=(a^6b^2c^6)/([a^4c^4+a^4(b^2-c^2)z^2+c^4(b^2-a^2)x^2]^2)
(15)
=(a^6b^6c^2)/([a^4b^4+b^4(c^2-a^2)x^2+a^4(c^2-b^2)y^2]^2).
(16)

The surface area of an ellipsoid is given by

S=2piabnsthetaint_0^theta((dn^2theta)/(dn^2u)+(cn^2theta)/(cn^2u))du
(17)
=2pi[c^2+(bc^2)/(sqrt(a^2-c^2))theta+bsqrt(a^2-c^2)E(am(theta),k)],
(18)

where ns(theta), dn(theta), and cn(theta) are Jacobi elliptic functions with modulus k,

k=(e_2)/(e_1)
(19)
e_1=sqrt((a^2-c^2)/(a^2))
(20)
e_2=sqrt((b^2-c^2)/(b^2)),
(21)

E(phi,k) is an incomplete elliptic integral of the second kind, am(phi) is the Jacobi amplitude with modulus k, and theta is given by inverting the expression

e_1=sn(theta,k),
(22)

where sn(theta) is another Jacobi elliptic function with modulus k (Bowman 1961, pp. 31-32; error corrected).

Another form of the surface area equation is

S=2pi[c^2+(bc^2)/(sqrt(a^2-c^2))F(phi,k)+bsqrt(a^2-c^2)E(phi,k)],
(23)

where

phi=sin^(-1)(sqrt(1-(c^2)/(a^2))).
(24)

The surface area can also be obtained directly from the first fundamental form as

S=int_0^piint_0^(2pi)sqrt(EG-F^2)dthetadphi
(25)
=int_0^pisinphiint_0^(2pi)sqrt(a^2b^2cos^2phi+c^2(b^2cos^2theta+a^2sin^2theta)sin^2phi)dthetadphi
(26)
=2sqrt(2)bint_0^pisqrt(a^2+c^2+(a^2-c^2)cos(2phi))sinphi×E(c/bsqrt((2(b^2-a^2))/(a^2+c^2+(a^2-c^2)cos(2phi)))sinphi)dphi.
(27)

A different parameterization of the ellipsoid is the so-called stereographic ellipsoid, given by the parametric equations

x(u,v)=(a(1-u^2-v^2))/(1+u^2+v^2)
(28)
y(u,v)=(2bu)/(1+u^2+v^2)
(29)
z(u,v)=(2cv)/(1+u^2+v^2).
(30)
EllipsoidMercator

A third parameterization is the Mercator parameterization

x(u,v)=asechvcosu
(31)
y(u,v)=bsechvsinu
(32)
z(u,v)=ctanhv
(33)

(Gray 1997).

The support function of the ellipsoid is

h=((x^2)/(a^4)+(y^2)/(b^4)+(z^2)/(c^4))^(-1/2),
(34)

and the Gaussian curvature is

K=(h^4)/(a^2b^2c^2)
(35)

(Gray 1997, p. 296).

The volume of the solid bounded by an ellipsoid with semi-axis lengths a,b,c is given by

V=4/3piabc.
(36)

The geometric centroids of the solid half-ellipsoids along the x-, y-, and z-axes are

x^_=3/(16)a
(37)
y^_=3/(16)b
(38)
z^_=3/(16)c.
(39)

The moment of inertia tensor of a solid ellipsoid is given by

I=[1/5M(b^2+c^2) 0 0; 0 1/5M(a^2+c^2) 0; 0 0 1/5M(a^2+b^2)].
(40)

See also

Confocal Ellipsoidal Coordinates, Confocal Quadrics, Convex Optimization Theory, Ellipsoid Packing, Goursat's Surface, Oblate Spheroid, Prolate Spheroid, Sphere, Spheroid, Superellipsoid

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 226, 1987.Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, 1961.Fischer, G. (Ed.). Plate 65 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 60, 1986.Gray, A. "The Ellipsoid" and "The Stereographic Ellipsoid." §13.2 and 13.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 301-303, 1997.Harris, J. W. and Stocker, H. "Ellipsoid." §4.10.1 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 111, 1998.Hilbert, D. and Cohn-Vossen, S. "The Thread Construction of the Ellipsoid, and Confocal Quadrics." §4 in Geometry and the Imagination. New York: Chelsea, pp. 19-25, 1999.JavaView. "Classic Surfaces from Differential Geometry: Ellipsoid." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Ellipsoid.html.Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, pp. 28 and 40-41, 1965.

Cite this as:

Weisstein, Eric W. "Ellipsoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ellipsoid.html

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