Conical surface
Surface drawn by a moving line passing through a fixed point
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Through Geometry, Surface (mathematics) and Line (geometry)
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 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.
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Special cases
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. 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.
Equations
A conical surface can be described parametrically as
- ,
where is the apex and 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 , 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.