Difference between revisions of "Cylindrical surface (cylinder)"
From Encyclopedia of Mathematics
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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 | ||
| − | + | $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1;$$ | |
an imaginary elliptic cylinder: | an imaginary elliptic cylinder: | ||
| − | + | $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=-1;$$ | |
a [[Hyperbolic cylinder|hyperbolic cylinder]]: | a [[Hyperbolic cylinder|hyperbolic cylinder]]: | ||
| − | + | $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1;$$ | |
and a [[Parabolic cylinder|parabolic cylinder]]: | and a [[Parabolic cylinder|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). | 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 | + | 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 | + | 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:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=-1;$$
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1;$$
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
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