Conical surface



In geometry, a conical surface is a three-dimensional surface formed from the union of lines that pass through a fixed point and a space curve.

Definitions
A (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point &mdash; the apex or vertex &mdash; and any point of some fixed space curve &mdash; the directrix &mdash; that does not contain the apex. Each of those lines is called a generatrix of the surface. The directrix is often taken as a plane curve, in a plane not containing the apex, but this is not a requirement.

In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve. Sometimes the term "conical surface" is used to mean just one nappe.

Special cases
If the directrix is a circle $$C$$, and the apex is located on the circle's axis (the line that contains the center of $$C$$ and is perpendicular to its plane), one obtains the right circular conical surface or double cone. More generally, when the directrix $$C$$ is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of $$C$$, one obtains an elliptic cone (also called a conical quadric or quadratic cone), which is a special case of a quadric surface.

Equations
A conical surface $$S$$ can be described parametrically as
 * $$S(t,u) = v + u q(t)$$,

where $$v$$ is the apex and $$q$$ is the directrix.

Related surface
Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points. Patches of conical surfaces that avoid the apex are special cases of developable surfaces, surfaces that can be unfolded to a flat plane without stretching. When the directrix has the property that the angle it subtends from the apex is exactly $$2\pi$$, then each nappe of the conical surface, including the apex, is a developable surface.

A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry a cylindrical surface is just a special case of a conical surface.