Hyperboloid

A non-closed central surface of the second order. One distinguishes between two types of hyperboloids: the one-sheet and the two-sheet hyperboloid.

Figure: h048360a

Figure: h048360b

In a suitable coordinate system (see Fig. a, Fig. b) the equation of a one-sheet hyperboloid is

while that of a two-sheet hyperboloid is

The numbers , and (and segments of such lengths) are known as the semi-axes of the hyperboloid. Sections of a hyperboloid by planes passing through the -axis are hyperbolas. Sections of a hyperboloid by planes perpendicular to the -axis are ellipses. The section of a one-sheet hyperboloid by the plane is said to be a gorge ellipse. A hyperboloid has three planes of symmetry. The cone defined by the equation

is called the asymptotic cone. If , the hyperboloid is said to be regular. A hyperboloid with two equal semi-axes is said to be a hyperboloid of rotation. A one-sheet hyperboloid is a ruled surface; the equations of the rectilinear generators passing through a given point have the form

References

[a1]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2]	D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)