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Difference between revisions of "Hyperboloid"

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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.
 
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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In a suitable coordinate system (see Fig. a, Fig. b) the equation of a [[One-sheet hyperboloid|one-sheet hyperboloid]] is
 
In a suitable coordinate system (see Fig. a, Fig. b) the equation of a [[One-sheet hyperboloid|one-sheet hyperboloid]] is
  
<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/h048360/h0483601.png" /></td> </tr></table>
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$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1,$$
  
 
while that of a [[Two-sheet hyperboloid|two-sheet hyperboloid]] is
 
while that of a [[Two-sheet hyperboloid|two-sheet hyperboloid]] is
  
<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/h048360/h0483602.png" /></td> </tr></table>
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$$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.$$
  
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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The numbers $a$, $b$ and $c$ (and segments of such lengths) are known as the semi-axes of the hyperboloid. Sections of a hyperboloid by planes passing through the $Oz$-axis are hyperbolas. Sections of a hyperboloid by planes perpendicular to the $Oz$-axis are ellipses. The section of a one-sheet hyperboloid by the plane $z=0$ is said to be a gorge ellipse. A hyperboloid has three planes of symmetry. The cone defined by the equation
  
<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/h048360/h0483609.png" /></td> </tr></table>
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$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$$
  
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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is called the asymptotic cone. If $a=b=c$, 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 $(x_0,y_0,z_0)$ 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/h048360/h04836012.png" /></td> </tr></table>
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$$\frac{x-x_0}{\frac{ay_0}b}=\frac{y-y_0}{\frac{-bx_0}a}=\frac{z-z_0}{c},$$
  
<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/h048360/h04836013.png" /></td> </tr></table>
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$$\frac{x-x_0}{\frac{-ay_0}b}=\frac{y-y_0}{\frac{bx_0}{a}}=\frac{z-z_0}{c}.$$
  
  

Latest revision as of 09:19, 22 August 2014

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

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1,$$

while that of a two-sheet hyperboloid is

$$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.$$

The numbers $a$, $b$ and $c$ (and segments of such lengths) are known as the semi-axes of the hyperboloid. Sections of a hyperboloid by planes passing through the $Oz$-axis are hyperbolas. Sections of a hyperboloid by planes perpendicular to the $Oz$-axis are ellipses. The section of a one-sheet hyperboloid by the plane $z=0$ is said to be a gorge ellipse. A hyperboloid has three planes of symmetry. The cone defined by the equation

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$$

is called the asymptotic cone. If $a=b=c$, 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 $(x_0,y_0,z_0)$ have the form

$$\frac{x-x_0}{\frac{ay_0}b}=\frac{y-y_0}{\frac{-bx_0}a}=\frac{z-z_0}{c},$$

$$\frac{x-x_0}{\frac{-ay_0}b}=\frac{y-y_0}{\frac{bx_0}{a}}=\frac{z-z_0}{c}.$$


Comments

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)
How to Cite This Entry:
Hyperboloid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperboloid&oldid=33079
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article