The surface formed by the movement of a straight line (the generator) through a given point (the vertex) intersecting a given curve (the directrix). A conical surface consists of two concave pieces positioned symmetrically about the vertex.
A second-order cone is one which has the form of a surface of the second order. The canonical equation of a real second-order conical surface is
if $a=b$, the surface is said to be circular or to be a conical surface of rotation; the canonical equation of an imaginary second-order canonical surface is
the only real point of an imaginary conical surface is $(0,0,0)$.
An $n$-th order cone is an algebraic surface given in affine coordinates $x,y,z$ by the equation
where $f(x,y,z)=0$ is a homogeneous polynomial of degree $n$ (a form of degree $n$ in $x,y,z$). If the point $M(x_0,y_0,z_0)$ lies on a cone, then the line $OM$ also lies on the cone ($O$ is the coordinate origin). The converse is also true: Every algebraic surface consisting of lines passing through a single point is a conical surface.
Conical surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conical_surface&oldid=31530