How to parametrize a cone?

Steps to Solve

1. Define the cone geometrically

A cone can be defined in a 3D coordinate system. For a cone with its vertex at the origin (0, 0, 0) and its axis pointing upward along the z-axis, we can describe its surface generally. The equation of a cone is given by:

$$ z = \sqrt{x^2 + y^2} $$

for a right circular cone.

2. Introduce parameters for the conic surface

To parametrize the cone, we use two parameters: an angle $\theta$ and a height $h$. We let:

$$ x = r \cos(\theta) $$ $$ y = r \sin(\theta) $$

Here, $r$ is the distance from the z-axis, which can be related to the height $h$ by the relationship $r = k \cdot h$, where $k$ is a constant that describes the slope of the cone.

3. Express the cone in parametric form

We can now combine these equations along with our earlier definition for $z$. Thus, the parametrization will be:

$$ \begin{align*} x &= k \cdot h \cdot \cos(\theta) \ y &= k \cdot h \cdot \sin(\theta) \ z &= h \end{align*} $$

where $\theta$ varies from $0$ to $2\pi$ and $h$ varies depending on the height you want, often from $0$ to some height $H$.

4. Summarizing the parameterization

The final parametrization is:

$$ \mathbf{p}(h, \theta) = \begin{bmatrix} k \cdot h \cdot \cos(\theta) \ k \cdot h \cdot \sin(\theta) \ h \end{bmatrix} $$

for $h \in [0, H]$ and $\theta \in [0, 2\pi]$, establishing a surface the forms a cone.