In geometry, a conical surface is an unbounded surface in three-dimensional space formed from the union of infinite 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 the apex or vertex and any point of some fixed space curve the directrix 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 rayss 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

, and the apex is located on the circle's axis and is perpendicular to its plane), one obtains the right circular conical surface or double cone. is an ellipse, or any conic section, one obtains an elliptic cone.

Equations

can be described parametrically as

1. ,

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, 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.

See also

• Conic section

• Quadric

References