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This chapter is motivated by the paper by Thurston on triangulations of thesphere and singular flat metrics on the sphere. Thurston locally parametrizedthe moduli space of singular flat metrics on the sphere with prescribedpositive curvature data by the complex hyperbolic space of appropriatedimension. This work can be considered as a generalization of signaturecalculation of the Hermitian form that he made in his paper.  The moduli space of singular flat metrics having unit area on the sphere withprescribed curvature data can be locally parametrized by certain spaces ofpolygons. This can be done by cutting singular flat spheres through lengthminimizing geodesics from a fixed singular point to the others. In that casethe space of polygons is a complex vector space of dimension $n-1$ when thereare $n+1$ singular points. Also there is natural area Hermitian form ofsignature $(1,n-2)$ on this vector space. In this chapter we calculate thesignature of the area Hermitian form on some spaces of polygons which locallyparametrize the moduli space of singular flat metrics having unit area on thesphere with one singular point of negative curvature. The formula we obtaindepends only on the sum of the curvatures of the singular points havingpositive curvature.  This paper will appear in the book "In the tradition of Thurston, Vol. II",Springer, 2022.

Signature Calculation of Area Hermitian Form on Some Spaces of Polygons