Parametric Equations for a Hypercone

Question

The n-dimensional cone, with vertex at the origin, central angle, $\alpha$ and central axis in the direction of the unit vector $\xi$ is defined to be all those points, $x\in {R^n}$ whose dot product with $\xi$ is |$x$|$cos(\alpha)$. How would I find parametric equations for this surface?

Answer 1

The easiest answer is if $\xi$ is along one of the axes, say the first. Then the cone can be parameterized as $(t\cos(\alpha ),t\sin(\alpha){\bf\hat{x}})$, where ${\bf\hat{x}}$ is a unit vector in $\mathbb{R^{n-1}}$. In coordinates, this is $(t\cos(\alpha ),t\sin(\alpha )\cos(\phi_1),t\sin(\alpha )\sin(\phi_1)\cos(\phi_2),t\sin(\alpha )\sin(\phi_1)\sin(\phi_2)\cos(\phi_3),\ldots)$ where you have n-1 angular terms and the last has no cosine. If n>3, $\phi_1$ runs from 0 to $\pi$ while the others run from 0 to $2\pi$. Then you can apply a rotation matrix to take $(1,0,0,\ldots)$ to $\xi$.