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Difference between revisions of "Cylindrical surface (cylinder)"

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The surface formed by the motion of a line (the generator) moving parallel to itself and intersecting a given curve (the directrix).
 
The surface formed by the motion of a line (the generator) moving parallel to itself and intersecting a given curve (the directrix).
  
 
The directrix of a cylindrical [[Surface of the second order|surface of the second order]] is a curve of the second order. Depending on the form of the directrix one distinguishes an [[Elliptic cylinder|elliptic cylinder]], the canonical equation of which is
 
The directrix of a cylindrical [[Surface of the second order|surface of the second order]] is a curve of the second order. Depending on the form of the directrix one distinguishes an [[Elliptic cylinder|elliptic cylinder]], the canonical equation of which 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/c/c027/c027650/c0276501.png" /></td> </tr></table>
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$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1;$$
  
 
an imaginary elliptic cylinder:
 
an imaginary elliptic cylinder:
  
<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/c/c027/c027650/c0276502.png" /></td> </tr></table>
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$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=-1;$$
  
 
a [[Hyperbolic cylinder|hyperbolic cylinder]]:
 
a [[Hyperbolic cylinder|hyperbolic cylinder]]:
  
<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/c/c027/c027650/c0276503.png" /></td> </tr></table>
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$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1;$$
  
 
and a [[Parabolic cylinder|parabolic cylinder]]:
 
and a [[Parabolic cylinder|parabolic cylinder]]:
  
<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/c/c027/c027650/c0276504.png" /></td> </tr></table>
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$$y^2=2px.$$
  
 
If the directrix is a degenerate curve of the second order (i.e. a pair of lines), then the cylindrical surface is a pair of planes (intersecting, parallel or coincident, real or imaginary, depending on the corresponding property of the directrix).
 
If the directrix is a degenerate curve of the second order (i.e. a pair of lines), then the cylindrical surface is a pair of planes (intersecting, parallel or coincident, real or imaginary, depending on the corresponding property of the directrix).
  
A cylindrical surface of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c0276506.png" /> is an algebraic surface given in some affine coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c0276507.png" /> by an algebraic equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c0276508.png" /> not containing one of the coordinates (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c0276509.png" />):
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A cylindrical surface of order $n$ is an algebraic surface given in some affine coordinate system $x,y,z$ by an algebraic equation of degree $n$ not containing one of the coordinates (for example, $z$):
  
<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/c/c027/c027650/c02765010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$f(x,y)=0.\tag{*}$$
  
The curve of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027650/c02765011.png" /> defined by equation (*) is sometimes called the base of the cylindrical surface.
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The curve of order $n$ defined by equation \ref{*} is sometimes called the base of the cylindrical surface.

Revision as of 12:17, 13 August 2014

The surface formed by the motion of a line (the generator) moving parallel to itself and intersecting a given curve (the directrix).

The directrix of a cylindrical surface of the second order is a curve of the second order. Depending on the form of the directrix one distinguishes an elliptic cylinder, the canonical equation of which is

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1;$$

an imaginary elliptic cylinder:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=-1;$$

a hyperbolic cylinder:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1;$$

and a parabolic cylinder:

$$y^2=2px.$$

If the directrix is a degenerate curve of the second order (i.e. a pair of lines), then the cylindrical surface is a pair of planes (intersecting, parallel or coincident, real or imaginary, depending on the corresponding property of the directrix).

A cylindrical surface of order $n$ is an algebraic surface given in some affine coordinate system $x,y,z$ by an algebraic equation of degree $n$ not containing one of the coordinates (for example, $z$):

$$f(x,y)=0.\tag{*}$$

The curve of order $n$ defined by equation \ref{*} is sometimes called the base of the cylindrical surface.

How to Cite This Entry:
Cylindrical surface (cylinder). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cylindrical_surface_(cylinder)&oldid=15188
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article