Difference between revisions of "Monge cone"
(TeX) |
m (label) |
||
| Line 3: | Line 3: | ||
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 | 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 | ||
| − | $$F(x,y,z,p,q)=0,\tag{*}$$ | + | $$F(x,y,z,p,q)=0,\label{*}\tag{*}$$ |
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. | 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. | ||
| Line 11: | Line 11: | ||
Figure: m064630a | Figure: m064630a | ||
| − | The geometric interpretation (see Fig.) of equation \ | + | The geometric interpretation (see Fig.) of equation \eqref{*}, as a field of directing cones, was given by G. Monge (1807). |
Latest revision as of 15:44, 14 February 2020
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
$$F(x,y,z,p,q)=0,\label{*}\tag{*}$$
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 \eqref{*}, as a field of directing cones, was given by G. Monge (1807).
Comments
See also Monge equation; Characteristic; Characteristic strip and the references therein.
Monge cone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monge_cone&oldid=32638