View source for Hyperbolic paraboloid

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<span id="Fig1">
[[File:Hyperbolic-paraboloid-1.gif| right| frame| Figure 1. A hyperbolic paraboloid]]
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A non-closed non-central [[surface of the second order]]. In a suitable coordinate system (see [[#Fig1|Figure&nbsp;1]]) the equation of a hyperbolic paraboloid is
\begin{equation}
\frac{x^2}{p}-\frac{y^2}{q}=2z, \qquad\text{where}\;p,q>0.
\end{equation}
Sections of a hyperbolic paraboloid by planes parallel to the planes $xOz$ and $yOz$ are [[parabola]]s, while sections by planes parallel to the plane $xOy$ are [[hyperbola]]s (the section by the plane $xOy$ consists of two straight lines). The symmetry axis of a hyperbolic paraboloid is said to be its axis; the point of intersection of a hyperbolic paraboloid with the axis is known as the apex. If $p=q$, the hyperbolic paraboloid has two axes of symmetry.

A hyperbolic paraboloid is a [[ruled surface]]; the equations of the rectilinear generators passing through a given point $(x_0,y_0,z_0)$ have the form
\begin{equation}
\begin{aligned}
\frac{x-x_0}{\sqrt{p}}=\frac{y-y_0}{\sqrt{q}}=\frac{z-z_0}{\frac{x_0}{\sqrt{p}}-\frac{y_0}{\sqrt{q}}}, \\
\frac{x-x_0}{\sqrt{p}}=\frac{y-y_0}{-\sqrt{q}}=\frac{z-z_0}{\frac{x_0}{\sqrt{p}}-\frac{y_0}{\sqrt{q}}}.
\end{aligned}
\end{equation}



====References====

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