Conical surface
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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.
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 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.
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.
See also
Conic section Quadric
References
Text derived from https://en.wikipedia.org/w/index.php?oldid=1378353555 under CC BY-SA 4.0; modifications have been made.