In geometry, a conical surface is an Unbounded set 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.
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 , surfaces that have a straight line through each of their points.
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
