**Up:** Part II: Three-Dimensional Geometry
** Previous:** 15 Surfaces of Revolution. The Torus

A surface defined by an algebraic equation of degree two is called a
**quadric**. Spheres, circular cylinders, and circular cones are
quadrics. By means of a rigid motion, any quadric can be transformed
into a quadric having one of the following equations (where *a*,*b*,*c*0):

(1) | Real ellipsoid | x/a+y/b+z/c=1 |

(2) | Imaginary ellipsoid | x/a+y/b+z/c=-1 |

(3) | Hyperboloid of one sheet | x/a+y/b-z/c=1 |

(4) | Hyperboloid of two sheets | x/a+y/b-z/c=-1 |

(5) | Real quadric cone | x/a+y/b-z/c=0 |

(6) | Imaginary quadric cone | x/a+y/b+z/c=0 |

(7) | Elliptic paraboloid | x/a+y/b+2z=0 |

(8) | Hyperbolic paraboloid | x/a-y/b+2z=0 |

(9) | Real elliptic cylinder | x/a+y/b=1 |

(10) | Imaginary elliptic cylinder | x/a+y/b=-1 |

(11) | Hyperbolic cylinder | x/a-y/b=1 |

(12) | Real intersecting planes | x/a-y/b=0 |

(13) | Imaginary intersecting planes | x/a+y/b=0 |

(14) | Parabolic cylinder |
x+2y=0 |

(15) | Real parallel planes |
x=1 |

(16) | Imaginary parallel planes |
x=-1 |

(17) | Coincident planes |
x=0 |

**Figure 1:** The ellipsoid
(1).

**Figure 2:** Left: hyperboloid of one sheet
(3).
Right: hyperboloid of two sheets
(4).

**Figure 3:** Left: elliptic paraboloid
(7).
Right: hyperbolic paraboloid
(8).

Surfaces with equations
(9)
--(17)
are
cylinders over the planes curves of the same equation
(Section 13.2). Equations
(2),
(6),
(10),
(16),
have no real solutions, so they do not describe surfaces in real
three-dimensional space. A surface
with equation (5)
can be
regarded as a cone (Section 13.3) over a conic *C* (any
ellipse, parabola or hyperbola can be taken as the directrix; there is
a two-parameter family of essentially distinct cones over it,
determined by the position of the vertex with respect to *C*).
The *real nondegenerate quadrics*
(1),
(3),
(4),
(7),
and
(8)
are shown in
Figures 1--3.

The surfaces with equations
(1)
--(6)
are **
central quadrics**; in the form given, the center is at the origin.
The quantities *a*, *b*, *c* are the **semiaxes**.

The **volume of the ellipsoid** with semiaxes *a*, *b*, *c* is
. When two of the semiaxes are the same, we can
also write the **area of the ellipsoid** in closed form. Suppose
*b=c*, so the ellipsoid *x*/*a*+(*y*+*z*)/*b*=1 is the surface of
revolution obtained by rotating the ellipse *x*/*a*+*y*/*b*=1 around
the *x*-axis. Its area is

The two quantities are equal, but only one avoids complex numbers,
depending on whether *a*>*b* or *a*<*b*. When *
a*>*b*, we have a **prolate spheroid**, that is, an ellipse
rotated around its major axis; when *a*<*b* we have an **
oblate spheroid**, which is an ellipse rotated around its minor axis.

Given a general quadratic equation in three variables,

*ax*+*by*+*cz*+2*fyz*+2*gzx*+2*hxy*+2*px*+2*qy*+2*rz*+*d*=0,

one can find out the type of conic it determines by consulting the following table:

k signs |
K signs |
Type of quadric | |||

3 | 4 | <0 | real ellipsoid | ||

3 | 4 | >0 | same | imaginary ellipsoid | |

3 | 4 | >0 | opposite | hyperboloid of one sheet | |

3 | 4 | <0 | opposite | hyperboloid of two sheets | |

3 | 3 | opposite | real quadric cone | ||

3 | 3 | same | imaginary quadric cone | ||

2 | 4 | <0 | same | elliptic paraboloid | |

2 | 4 | >0 | opposite | hyperbolic paraboloid | |

2 | 3 | same | opposite | real elliptic cylinder | |

2 | 3 | same | same | imaginary elliptic cylinder | |

2 | 3 | opposite | hyperbolic cylinder | ||

2 | 2 | opposite | real intersecting planes | ||

2 | 2 | same | imaginary intersecting planes | ||

1 | 3 | parabolic cylinder | |||

1 | 2 | opposite | real parallel planes | ||

1 | 2 | same | imaginary parallel planes | ||

1 | 1 | coincident planes |

The columns have the following meaning. Let

let and be the ranks of *e* and *E*, and let
be the determinant of *E*. The column ``*k* signs'' refers
to the nonzero eigenvalues of *e*, that is, the roots of

if all nonzero eigenvalues have the same sign, choose ``same'',
otherwise ``opposite''. Similarly, ``*K* signs'' refers to the sign
of the nonzero eigenvalues of *E*.

**Up:** Part II: Three-Dimensional Geometry
** Previous:** 15 Surfaces of Revolution. The Torus

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.