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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 $E=5,E=6,...,E\approx20$.

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    $\begingroup$ Consider a spherical rhombus R. Why should not two polygons differ in area by the area of R, and still share the same configuration of sides (excepting those sides involved in R)? I do not see how you can expect the area to depend only on the number of sides, even given all sides are the same length. Gerhard "Ask Me About System Design" Paseman, 2012.05.22 $\endgroup$
    – Gerhard Paseman
    May 23, 2012 at 4:04
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    $\begingroup$ The area of a simple $n$-gon on the unit sphere is $(n-2)\pi - \sum_i \gamma_i$, where $\gamma_i$ is the $i$th internal angle. Since you know $\sum_i \gamma_i$, you're done. (I don't understand why the discussion at Wikipedia was so messy; the area of a regular spherical 4-gon with internal angle $\gamma$ is $2\pi-4\gamma$.) $\endgroup$
    – JeffE
    May 23, 2012 at 7:46

3 Answers 3

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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

  • smh
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You have to be careful with the definition of spherical polygon. I suppose you mean a polygon whose edges are geodesiscs on the sphere, i.e., arcs of great circles. In this case, the Gauss-Bonnet formula gives you the answer entirely in terms of the outer angles of the polygon. The lengths of the edges is irrelevant in this case.

If the edges are not geodesics, the Gauss-Bonnet formula still gives you an answer, but you need to know the geodesic curvatures of the edges. (When edges are geodesics, this geodesic curvature vanishes.)

In the case of (geodesic) polygons on the sphere there are elementary proofs of the Gauss-Bonnet formula that do not require any sophisticated knowledge of differential geometry. I refer you to theses elementary notes for a talk I gave to first year grad students a while ago.

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My library GeographicLib computes the area of geodesic polygons on an ellipsoid of revolution. Spherical areas are just a special case of this. The library includes a Planimeter utility which computes the area given the vertices. Just specify a unit sphere with "-e 1 0".

The method used is described in my paper, Algorithms for geodesics, section 6. Of course, the algorithm simplifies considerably in the spherical case.

There are 2 key points:

  1. The use of Bessel's formula, Eq. (64) in the paper, to compute the area contributed by a single polygon edge; this is a spherical generalization of the familiar trapezium formula for planar poygons. Bessel's formula is considerably better in this application than l'Huilier's formula (advocated by Chamberlain and Duquette), which gives the area of a triangle in terms of its sides; this yields inaccurate results with ill-conditional triangles.

  2. You need to account for encircling the pole.

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