Conical surface
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In geometry, 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.
In general, a conical surface consists of two identical unbounded halves joined by the vertex. However, in some cases, these two halves may intersect, or even coincide.
In particular, 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. This special case is often called a cone, because it is one of the two distinct surfaces that bound the geometric solid of that name. This geometric object can also be described as the set of all points swept by a line that intercepts a the axis line and rotates around it; or the union of all lines that intersect the axis at a fixed point <math>p<math>and at a fixed angle <math>\theta<math>. The aperture of the cone is the angle <math>2 \theta<math>.
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 a conical quadric, which is a special case of a quadric — indeed a ruled and developable one.
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 there is no difference between the cylindrical and conical surfaces, and the two halves of the latter become a single connected surface.
Equations
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.
A right circular cone whose axis is the <math>Z<math> coordinate axis, and whose apex is the origin, it is described parametrically as
- <math>S(t,u) = (u \cos\theta \cos t, u \cos\theta \sin t, u \sin\theta)<math>
and in implicit form by <math>S(x,y,z) = 0<math> where
- <math>S(x,y,z) = (x^2 + y^2)(\cos\theta)^2 - z^2 (\sin \theta)^2.<math>
More generally, a right circular cone with vertex at the origin, axis parallel to the vector <math>d<math>, and aperture <math>2\theta<math>, is given by the implicit vector equation <math>S(u) = 0<math> where
- <math>S(u) = (u . d)^2 - (d . d) (u . u) (\cos \theta)^2<math>
or
- <math>S(u) = u . d - |d| |u| \cos \theta<math>
where <math>u=(x,y,z)<math>, and <math>u.d<math> denotes the dot product.