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.

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.^[1]

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.^[2] Sometimes the term "conical surface" is used to mean just one nappe.^[3]

If the directrix is a circle ${\displaystyle C}$, and the apex is located on the circle's axis (the line that contains the center of ${\displaystyle C}$ and is perpendicular to its plane), one obtains the right circular conical surface or double cone.^[2] More generally, when the directrix ${\displaystyle C}$ is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of ${\displaystyle C}$, one obtains an elliptic cone^[4] (also called a conical quadric or quadratic cone),^[5] which is a special case of a quadric surface.^[4][5]

A conical surface ${\displaystyle S}$ can be described parametrically as

${\displaystyle S(t,u)=v+uq(t)}$,

where ${\displaystyle v}$ is the apex and ${\displaystyle q}$ is the directrix.^[6]

Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points.^[7] 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 ${\displaystyle 2\pi }$, then each nappe of the conical surface, including the apex, is a developable surface.^[8]

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.^[9]

• Conic section
• Quadric