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A non-closed non-central [[Surface of the second order|surface of the second order]]. In a suitable coordinate system (see Fig.) the equation of a hyperbolic paraboloid is
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048290/h0482901.png" /></td> </tr></table>
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<span id="Fig1">
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[[File:Hyperbolic-paraboloid-1.gif| right| frame| Figure 1. A hyperbolic paraboloid]]
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</span>
  
Sections of a hyperbolic paraboloid by planes parallel to the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048290/h0482902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048290/h0482903.png" /> are parabolas, while sections by planes parallel to the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048290/h0482904.png" /> are hyperbolas (the section by the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048290/h0482905.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048290/h0482906.png" />, the hyperbolic paraboloid has two axes of symmetry.
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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
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\begin{equation}
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\frac{x^2}{p}-\frac{y^2}{q}=2z, \qquad\text{where}\;p,q>0.
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\end{equation}
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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.
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h048290a.gif" />
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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
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\begin{equation}
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\begin{aligned}
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\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}}}, \\
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\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}}}.
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\end{aligned}
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\end{equation}
  
Figure: h048290a
 
  
A hyperbolic paraboloid 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/h048290/h0482907.png" /> have the form
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048290/h0482908.png" /></td> </tr></table>
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048290/h0482909.png" /></td> </tr></table>
 
  
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====References====
  
 
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<table>
====Comments====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR>
 
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<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>
 
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</table>
====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>
 

Latest revision as of 11:12, 25 May 2016


Figure 1. A hyperbolic paraboloid

A non-closed non-central surface of the second order. In a suitable coordinate system (see Figure 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 parabolas, while sections by planes parallel to the plane $xOy$ are hyperbolas (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

[1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[2] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)
How to Cite This Entry:
Hyperbolic paraboloid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_paraboloid&oldid=16696
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article