:PROPERTIES: :ID: 63cbda97-30c2-4bf2-8d74-137580e203f6 :mtime: 20231014005637 20231010015154 :ctime: 20231010015019 :END: #+title: equation of a cone #+filetags: :public:project: * Definition Given a vertex $Q$ which has position vector $\vec{q}$, and given that the cone has semi-angle $\alpha$, and given that the axis of the cone is parallel to [[id:546e7943-327d-41b8-8b35-37053964595b][unit vector]] $\hat{n}$, then all points on the cone can be generated by position vectors $\vec{r}$ such that: $(\vec{r} - \vec{q}) \cdot \hat{n} / |\vec{r} - \vec{q} = \cos \alpha$