The Amsler Surface: Complete Theory and Numerical Implementation

This comprehensive document presents the complete mathematical theory of the Amsler surface — a classical pseudospherical surface of constant negative curvature K = −1 — for the general case where the initial angle ω0 between the straight asymptotic lines is arbitrary. We provide a complete derivation of the differential equations, including the correct initial conditions that have been a source of confusion in the literature. Last version is available at https://zenodo.org/records/18253019