Conical surface
cone
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
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if , 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
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the only real point of an imaginary conical surface is .
An -th order cone is an algebraic surface given in affine coordinates
by the equation
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where is a homogeneous polynomial of degree
(a form of degree
in
). If the point
lies on a cone, then the line
also lies on the cone (
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