Ellipsoid

A Quadratic Surface which is given in Cartesian Coordinates by

$${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1,$$ (1)

where the semi-axes are of lengths $a$, $b$, and $c$. In Spherical Coordinates, this becomes

$${r^2\cos^2\theta\sin^2\phi\over a^2}+{r^2\sin^2\theta\sin^2\phi\over b^2} + {r^2\cos^2\phi\over c^2} = 1.$$ (2)

The parametric equations are

$$x = a\cos\theta\sin\phi$$ (3)

$$y = b\sin\theta\sin\phi$$ (4)

$$z = c\cos\phi.$$ (5)

The Surface Area (Bowman 1961, pp. 31-32) is

$$S=2\pi c^2+{2\pi b\over\sqrt{a^2-c^2}} [(a^2-c^2)E(\theta)+c^2\theta],$$ (6)

where $E(\theta)$ is a Complete Elliptic Integral of the Second Kind,

$${e_1}^2 \equiv {a^2-c^2\over a^2}$$ (7)

$${e_2}^2 \equiv {b^2-c^2\over b^2}$$ (8)

$$k \equiv {e_2\over a_1},$$ (9)

and $\theta$ is given by inverting the expression

$$e_1=\mathop{\rm sn}\nolimits (\theta,k),$$ (10)

where $\mathop{\rm sn}\nolimits (\theta,k)$ is a Jacobi Elliptic Function. The Volume of an ellipsoid is

$$V ={\textstyle{4\over 3}}\pi abc.$$ (11)

If two axes are the same, the figure is called a Spheroid (depending on whether $c<a$ or $c>a$, an Oblate Spheroid or Prolate Spheroid, respectively), and if all three are the same, it is a Sphere.

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

$$x(u,v) = {a(1-u^2-v^2)\over 1+u^2+v^2}$$ (12)

$$y(u,v) = {2bu\over 1+u^2+v^2}$$ (13)

$$z(u,v) = {2cv\over 1+u^2+v^2}.$$ (14)

A third parameterization is the Mercator parameterization

$$x(u,v) = a\mathop{\rm sech}\nolimits v\cos u$$ (15)

$$y(u,v) = b\mathop{\rm sech}\nolimits v\sin u$$ (16)

$$z(u,v) = c\tanh v$$ (17)

(Gray 1993).

The Support Function of the ellipsoid is

$$h=\left({{x^2\over a^4}+{y^2\over b^4}+{z^2\over c^4}}\right)^{-1/2},$$ (18)

and the Gaussian Curvature is

$$K={h^4\over a^2b^2c^2}$$ (19)

(Gray 1993, p. 296).

See also Convex Optimization Theory, Oblate Spheroid, Prolate Spheroid, Sphere, Spheroid

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 131, 1987.

Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, 1961.

Fischer, G. (Ed.). Plate 65 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 60, 1986.

Gray, A. ``The Ellipsoid'' and ``The Stereographic Ellipsoid.'' §11.2 and 11.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 215-217, and 296, 1993.