In geometry, a conical surface is an unbounded three-dimensional surface formed from the union of infinite lines that pass through a fixed point and a space curve.
DefinitionsEdit
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.<ref>Template:Citation</ref>
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.<ref name=msg>Template:Citation</ref> Sometimes the term "conical surface" is used to mean just one nappe.<ref>Template:Citation</ref>
Special casesEdit
If the directrix is a circle <math>C</math>, and the apex is located on the circle's axis (the line that contains the center of <math>C</math> and is perpendicular to its plane), one obtains the right circular conical surface or double cone.<ref name=msg/> More generally, when the directrix <math>C</math> is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of <math>C</math>, one obtains an elliptic cone<ref name=young/> (also called a conical quadric or quadratic cone),<ref name=osg>Template:Citation</ref> which is a special case of a quadric surface.<ref name=young>Template:Citation</ref><ref name=osg/>
EquationsEdit
A conical surface <math>S</math> can be described parametrically as
- <math>S(t,u) = v + u q(t)</math>,
where <math>v</math> is the apex and <math>q</math> is the directrix.<ref>Template:Citation</ref>
Related surfaceEdit
Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points.<ref>Template:Citation</ref> 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 <math>2\pi</math>, then each nappe of the conical surface, including the apex, is a developable surface.<ref>Template:Citation</ref>
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.<ref>Template:Citation</ref>