Editing Cone (section)

====Equation form====

The surface of a cone can be parameterized as
:<math>f(\theta,h) = (h \cos\theta, h \sin\theta, h ),</math>
where <math>\theta \in [0,2\pi)</math> is the angle "around" the cone, and <math>h \in \mathbb{R}</math> is the "height" along the cone.

A right solid circular cone with height <math>h</math> and aperture  <math>2\theta</math>, whose axis is the <math>z</math> coordinate axis and whose apex is the origin, is described parametrically as
:<math>F(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right)</math>
where <math>s,t,u</math> range over <math>[0,\theta)</math>, <math>[0,2\pi)</math>, and <math>[0,h]</math>, respectively.

In [[Implicit function|implicit]] form, the same solid is defined by the inequalities
:<math>\{ F(x,y,z) \leq 0, z\geq 0, z\leq h\},</math>
where
:<math>F(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 calculus|vector]] equation <math>F(u) = 0</math> where
:<math>F(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2</math>
:<math>F(u) = u \cdot d - |d| |u| \cos \theta</math>
where <math>u=(x,y,z)</math>, and <math>u \cdot d</math> denotes the [[dot product]].