EPH-classifications in Geometry, Algebra, Analysis and Arithmetic

Trichotomy of Elliptic-Parabolic-Hyperbolic appears in many different areas of mathematics. All of these are named after the very first example of trichotomy, which is formed by ellipses, parabolas, and hyperbolas as conic sections. We try to understand if these classifications are justified and if similar mathematical phenomena is shared among different cases EPH-classification is used.