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Difference between revisions of "Monge cone"

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''directing cone''
 
''directing cone''
  
The [[Envelope|envelope]] of the tangent planes to the integral surface at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m0646301.png" /> of a partial differential equation
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The [[Envelope|envelope]] of the tangent planes to the integral surface at a point $(x_0,y_0,z_0)$ of a partial differential 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/m/m064/m064630/m0646302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$F(x,y,z,p,q)=0,\tag{*}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m0646303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m0646304.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m0646305.png" /> is a non-linear function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m0646306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m0646307.png" />, then the general case holds: The tangent planes form a one-parameter family of planes passing through a fixed point; their envelope is a cone. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m0646308.png" /> is a linear function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m0646309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m06463010.png" />, then a bundle of planes passing through a line is obtained, that is, the Monge cone degenerates to the so-called Monge axis. The directions of the generators of the Monge cone corresponding to some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064630/m06463011.png" /> are called characteristic directions. A line on the integral surface which is tangent at each point to a corresponding generator of the Monge cone is called a characteristic line, a characteristic, a focal curve, or a Monge curve.
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where $p=\partial z/\partial x$, $q=\partial z/\partial y$. If $F$ is a non-linear function in $p$ and $q$, then the general case holds: The tangent planes form a one-parameter family of planes passing through a fixed point; their envelope is a cone. If $F$ is a linear function in $p$ and $q$, then a bundle of planes passing through a line is obtained, that is, the Monge cone degenerates to the so-called Monge axis. The directions of the generators of the Monge cone corresponding to some point $(x_0,y_0,z_0)$ are called characteristic directions. A line on the integral surface which is tangent at each point to a corresponding generator of the Monge cone is called a characteristic line, a characteristic, a focal curve, or a Monge curve.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/m064630a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/m064630a.gif" />
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Figure: m064630a
 
Figure: m064630a
  
The geometric interpretation (see Fig.) of equation (*), as a field of directing cones, was given by G. Monge (1807).
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The geometric interpretation (see Fig.) of equation \ref{*}, as a field of directing cones, was given by G. Monge (1807).
  
  

Revision as of 08:58, 1 August 2014

directing cone

The envelope of the tangent planes to the integral surface at a point $(x_0,y_0,z_0)$ of a partial differential equation

$$F(x,y,z,p,q)=0,\tag{*}$$

where $p=\partial z/\partial x$, $q=\partial z/\partial y$. If $F$ is a non-linear function in $p$ and $q$, then the general case holds: The tangent planes form a one-parameter family of planes passing through a fixed point; their envelope is a cone. If $F$ is a linear function in $p$ and $q$, then a bundle of planes passing through a line is obtained, that is, the Monge cone degenerates to the so-called Monge axis. The directions of the generators of the Monge cone corresponding to some point $(x_0,y_0,z_0)$ are called characteristic directions. A line on the integral surface which is tangent at each point to a corresponding generator of the Monge cone is called a characteristic line, a characteristic, a focal curve, or a Monge curve.

Figure: m064630a

The geometric interpretation (see Fig.) of equation \ref{*}, as a field of directing cones, was given by G. Monge (1807).


Comments

See also Monge equation; Characteristic; Characteristic strip and the references therein.

How to Cite This Entry:
Monge cone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monge_cone&oldid=15320
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article