Short version: if two spherical caps of the same sphere intersect, how can I determined the coordinates of the two "singular points" of this intersection
Long version: On a unit sphere, centered at the original, we consider two spherical cap. Each cap is defined by:
- the (unit) vector indicating the top of the spherical cap, $\mathbf{n}$
- the angle $\theta$, defined as is done there; any point $\mathbf{r}$ on the sphere belongs to the cap iff $\mathrm{angle}(\mathbf{n},\mathbf{r}) \le \theta$
The question is: given two spherical caps $(\mathbf{n}_1,\theta_1)$ and $(\mathbf{n}_2,\theta_2)$, their intersection can be:
- empty
- a single point
- a surface delimited by two circular arcs, intersecting in two points
The question is: in the third case, how can I determine the coordinates of the two points of interest? I'm sure it's not so hard if you find the right way to go at it.