View source for Hyperboloid

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

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Figure: h048360a

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Figure: h048360b

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

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while that of a [[Two-sheet hyperboloid|two-sheet hyperboloid]] is

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The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048360/h0483603.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048360/h0483604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048360/h0483605.png" /> (and segments of such lengths) are known as the semi-axes of the hyperboloid. Sections of a hyperboloid by planes passing through the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048360/h0483606.png" />-axis are hyperbolas. Sections of a hyperboloid by planes perpendicular to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048360/h0483607.png" />-axis are ellipses. The section of a one-sheet hyperboloid by the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048360/h0483608.png" /> is said to be a gorge ellipse. A hyperboloid has three planes of symmetry. The cone defined by the equation

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is called the asymptotic cone. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048360/h04836010.png" />, 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|ruled surface]]; the equations of the rectilinear generators passing through a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048360/h04836011.png" /> have the form

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====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Hilbert,   S.E. Cohn-Vossen,   "Geometry and the imagination" , Chelsea  (1952)  (Translated from German)</TD></TR></table>