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.
Best Answer
Put ${\bf n}:={\bf n}_1\times{\bf n}_2/|{\bf n}_1\times{\bf n}_2|$ and write ${\bf x}:=\xi_1{\bf n}_1+ \xi_2 {\bf n_2} + \xi_3 {\bf n}$ with unknown $\xi_i$. Then $$\cos\theta_1={\bf x}\bullet {\bf n}_1=\xi_1 +\xi_2({\bf n}_1\bullet{\bf n}_2)$$ and similarly $\cos\theta_2=\ldots\ $. After $\xi_1$ and $\xi_2$ have been computed the value of $\xi_3$ follows from the condition $|{\bf x}|=1$.