I need some sort of classification (up to isometry) of spherical polygons (i.e. polygons in $\mathbb{S}^2$ whose edges are given by geodesics) subject to the interior angles and the perimeter of the polygon itself. Most likely this question is already solved and can be found in some literature, but I couldn't find something in this direction. Does anyone know where I can find something related?


Most of the literature just deal with triangles but do not go further and study polygons with more than three vertices (not even a word is said about it). In case of triangles the situation is very clear! The interior angles determine a spherical triangle up to isometry. But what about more than three vertices?


Best regards



Classification of spherical polygons
You should specify what you mean by a polygon: a broken line or a surface.
If we are talking about a broken line, then some classification is given here:


MR1703691 Kapovich, Michael; Millson, John J. On the moduli space of a spherical polygonal linkage. Canad. Math. Bull. 42 (1999), no. 3, 307–320.


If we are talking about surfaces, this is a different problem which is much more complicated.
When the angles are sufficiently small, so that the whole polygon is a subset of the sphere,
a classification is obtained in


F. Luo and G. Tian, Liouville equation and spherical convex polytopes, Proc. AMS, 116 (1992) 4, 1119-1129.


If the interior angles can be arbitrarily large, there is no known classification, even for the case of quadrilaterals. Some partial results can be found in 


Eremenko, Gabrielov, Tarasov, arXiv:1405.1738 
Metrics with conic singularities and spherical polygons


We actually have a classification of quadrilaterals up to isometry but it is not ready for publication yet.


To answer your more specific question. A spherical polygon generally depends on 2n-3 real
parameters: $n$ angles and $n-3$ additional parameters. For these additional parameters,
one can take certain conformal moduli, for example accessory parameters in the differential equation which is associated to this polygon. So for $n\geq 4$, we have 5 parameters, but whether there exist two quadrilaterals with the same angles and perimeter we did not
investigate. For $n\geq 5$ polygons with the same angles and perimeter certainly exist,
by dimension count.


EDIT. That $n$-gons with the same angles and same perimeters exist, can be obtained from our other paper, arXiv:math/0405196 where polygons whose angles are integer multiples of $2\pi$ are classified.


