structural:expansion, information:extraction
Equation form

The surface of a cone can be parameterized as

where θ ∈ [ 0, 2 π) {\displaystyle \theta \in [0,2\pi)} is the angle "around" the cone, and h ∈ R {\displaystyle h\in \mathbb {R} } is the "height" along the cone.

A right solid circular cone with height h {\displaystyle h} and aperture 2 θ {\displaystyle 2\theta }, whose axis is the z {\displaystyle z} coordinate axis and whose apex is the origin, is described parametrically as

where s, t, u {\displaystyle s,t,u} range over {\displaystyle }, respectively.

In implicit form, the same solid is defined by the inequalities

where

More generally, a right circular cone with vertex at the origin, axis parallel to the vector d {\displaystyle d}, and aperture 2 θ {\displaystyle 2\theta }, is given by the implicit vector equation F (u) = 0 {\displaystyle F(u)=0} where

where u = (x, y, z) {\displaystyle u=(x,y,z)}, and u ⋅ d {\displaystyle u\cdot d} denotes the dot product.