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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,
\end{displaymath} (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.
\end{displaymath} (2)

The parametric equations are
$\displaystyle x$ $\textstyle =$ $\displaystyle a\cos\theta\sin\phi$ (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle b\sin\theta\sin\phi$ (4)
$\displaystyle z$ $\textstyle =$ $\displaystyle c\cos\phi.$ (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],
\end{displaymath} (6)

where $E(\theta)$ is a Complete Elliptic Integral of the Second Kind,
$\displaystyle {e_1}^2$ $\textstyle \equiv$ $\displaystyle {a^2-c^2\over a^2}$ (7)
$\displaystyle {e_2}^2$ $\textstyle \equiv$ $\displaystyle {b^2-c^2\over b^2}$ (8)
$\displaystyle k$ $\textstyle \equiv$ $\displaystyle {e_2\over a_1},$ (9)

and $\theta$ is given by inverting the expression
e_1=\mathop{\rm sn}\nolimits (\theta,k),
\end{displaymath} (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.
\end{displaymath} (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

$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle {a(1-u^2-v^2)\over 1+u^2+v^2}$ (12)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle {2bu\over 1+u^2+v^2}$ (13)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle {2cv\over 1+u^2+v^2}.$ (14)

A third parameterization is the Mercator parameterization

$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle a\mathop{\rm sech}\nolimits v\cos u$ (15)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle b\mathop{\rm sech}\nolimits v\sin u$ (16)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle c\tanh v$ (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},
\end{displaymath} (18)

and the Gaussian Curvature is
K={h^4\over a^2b^2c^2}
\end{displaymath} (19)

(Gray 1993, p. 296).

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


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.

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© 1996-9 Eric W. Weisstein