Context:
On a sphere, I'm given a positively oriented spherical polygon whose vertices are (in order) $$ A(R,\phi_1,\theta_1),\ B(R,\phi_2,\theta_2),\ C(R,\phi_3,\theta_3),\ \cdots, (R,\phi_n,\theta_n).$$ How can I calculate the area of the spherical polygon?
I know that the area $S$ of spherical polygon is tightly related to the spherical excess and therefore $$ S = R^2\left[\sum_{i=1}^{n}\alpha_i - (n-2)\pi\right]. $$
Question: How could I obtain the interior angles $\alpha_i$ of the spherical polygon?
As an illustration, the three vertices $A$, $B$ and $C$ will contribute to $\alpha_2$ which is the interior angle at vertex $B(R,\phi_2,\theta_2)$.
Since the length of arcs $\newcommand{arc}[1]{\stackrel{\Large\frown}{#1}}\arc{AB}$, $\arc{BC}$ and $\arc{CA}$ could be calculated easily from the great circle distances, I think I have a good starting point to calculate the interior angle $B$ because
- I have three sides of the spherical triangle $\Delta ABC$, which means that
- the spherical angle at $B$ in spherical triangle $\Delta ABC$ is uniquely determined.
However, I don't really know how to solve for the spherical angle given three sides.
To make matters worse, solving the spherical triangle doesn't actually tell me whether I should pick $\alpha_2=B$ or $\alpha_2=2\pi-B$ as the interior angle. As seen in the illustrated case, the reflex angle $2\pi-B$ is the interior angle.
Answers that can contribute to my learning could calculate the interior angle from the list of spherical coordinates
- by solving the spherical triangle and then account for the concave / convex angle,
- by illustrating the use of vectors, or
- from an entirely new perspective.