Monge cone

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).