Ellipsoid

(real)

2020 Mathematics Subject Classification: Primary: 51-XX [MSN][ZBL]

Figure 1. An ellipsoid with $a$, $b$, $c$ in the ratio 8:3:6 (pdf)

An ellipsoid is a closed central surface of the second order (see Figure 1).

The canonical equation of an ellipsoid has the form $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$ The positive numbers $a$, $b$ and $c$ and the segments of corresponding lengths are called the semi-axes of the ellipsoid. The section of an ellipsoid by any plane is an ellipse.

If two semi-axes of an ellipsoid are equal, the ellipsoid is called an ellipsoid of revolution, and the sections of an ellipsoid of revolution by planes parallel to the plane of the equal semi-axes are circles. When $a=b=c$, the ellipsoid is a sphere. The centre of symmetry of an ellipsoid is called the centre of the ellipsoid.

A second-order surface with canonical equation $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$ is called an imaginary ellipsoid.

Comments

As the name suggests, an imaginary ellipsoid does not have real points.

Other characterizations of an ellipsoid are as follows: An ellipsoid is the affine image of a sphere; an ellipsoid is a non-degenerate quadric without real points at infinity.

A systematic treatment of ellipsoids is given in [Be], Chapt. 15 and [Co]. A more advanced survey is contained in [Pe].

References

[Be]	M. Berger, "Geometry", II, Springer (1987) MR0882916 Zbl 0606.51001
[Co]	J. Coolidge, "A history of the conic sections and quadric surfaces", Dover, reprint (1968) MR0245397 Zbl 0060.01006
[Pe]	C.M. Petty, "Ellipsoids" P.M. Gruber (ed.) J.M. Wills (ed.), Convexity and its applications, Birkhäuser (1983) pp. 264–276 MR0731103 Zbl 0512.00020