Parametric Equations for a Hypercone

The n - dimensional cone, with vertex at the beginning, main angle, $\alpha$ and also main axis towards the device vector $\xi$ is specified to be all those factors, $x\in {R^n}$ whose dot item with $\xi$ is |$x$ |$cos(\alpha)$. Just how would certainly I locate parametric formulas for this surface area?

2019-12-02 02:51:26
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The most convenient solution is if $\xi$ is along among the axes, claim the first. After that the cone can be parameterized as $(t\cos(\alpha ),t\sin(\alpha){\bf\hat{x}})$, where ${\bf\hat{x}}$ is a device vector in $\mathbb{R^{n-1}}$. In works with, 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 also the last has no cosine. If n > 3, $\phi_1$ runs from 0 to $\pi$ while the others range from 0 to $2\pi$. After that you can use a turning matrix to take $(1,0,0,\ldots)$ to $\xi$.

2019-12-03 04:13:57