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 , and the apex is located on the circle's *axis* (the line that contains the center of and is perpendicular to its plane), one obtains the *right circular conical surface* or double cone.^{[2]} More generally, when the directrix is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of , 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 can be described parametrically as

- ,

where is the apex and 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 , 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]}