Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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:

  1. empty
  2. a single point
  3. 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.

share|improve this question
    
Related... wait, that's another question of yours. –  J. M. Dec 12 '10 at 12:13
add comment

1 Answer 1

up vote 1 down vote accepted

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 $|{\bf x}|=1$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.