Conical surface

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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.

How to Cite This Entry:
Conical surface. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article