Classification of generic spherical quadrilaterals

TL;DR

This paper classifies generic spherical quadrilaterals up to isometry, showing that with prescribed angles, their space consists of finitely many open curves, and describes degeneration at endpoints.

Contribution

It provides a classification of generic spherical quadrilaterals based on their angles and develops a detailed description of their moduli space and degenerations.

Findings

The space of quadrilaterals with fixed angles forms finitely many open curves.

Degeneration behavior at the endpoints of these curves is characterized.

Conditions for genericity involve images of sides lying on four distinct circles with no triple intersections.

Abstract

Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined.