Conical surface: Difference between revisions

Revision as of 20:18, 17 August 2006

Revision as of 18:57, 22 August 2006

Line 4:

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. Every conic surface is [[ruled surface|ruled]] and [[developable]].

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 (solidgeometry)|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 [[rotation|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]].