I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can assume a radius of $R=1$).
For what I am researching (I will not go into the background) I need the following conditions to be satisfied:
- The edge length of all spherical polygons in $\mathbb{S}^2$ is $\pi / 3$.
- The spherical polygons I want to consider may or may not be convex, in fact it is necessary that I be able to compute the area of a non-convex polygon in $\mathbb{S}^2$.
I know there are multiple ways to compute the area of a spherical triangle using the spherical law of cosines, L'Huilier's theorem, or other formulas, but I want to be more general than this. The type of information I know about the internal angles of the spherical polygons is a bound in terms of the degree of the vertex considered in a spherical simplicial $2$-complex $\mathcal{K}$ in $\mathbb{S}^2$. That is, labeling $\gamma_{i}$ as an internal angle of the spherical polygon that
$$ \sum\limits_{1 \leq i \leq b} \gamma_{i} = \sum\limits_{1 \leq i \leq b} (i-1)\arccos(\frac{1}{3})b_{i}$$
where $b_{i}$ denotes the number of vertices of degree $i$ in $\mathcal{K}$. The last comment about the internal angles may or may not be confusing, but I just wanted to mention that I know something about the internal angles of the spherical polygons. For an example of how difficult this problem may be, there was a large discussion here about determining the area of a spherical $4$-gon with given side length (and the answer was quite messy), so I'm hoping that some of you have ideas!
To summarize exactly what my question is, and what information I know:
You are given a number $E$ which tells you how many edges a spherical polygon $C$ in $\mathbb{S}^2$ has (all edges have length $\pi /3$ and $C$ is not necessarily convex). Determine the area of $C$ (or a function for the area of $C$).
That is, I want to find the analogue in spherical geometry to the equations in Euclidean geometry which tell you the area of a regular polygon of a given number of sides. If such a general expression does not exist, I would be interested in the case for $E=5,E=6,…,E\approx20$.
Best Answer
This excellent paper collects many useful formulae for spherical calculations, including (but hardly limited to) polygon area. Explanations are clear and well developed.
Some Algorithms for Polygons on a Sphere Robert.G.Chamberlain William.H.Duquette Jet Propulsion Laboratory
http://hdl.handle.net/2014/40409