Solving the Bonnet problem - A hands-on adventure in 17 chapters
This page showcases the animations produced for the movie Solving the Bonnet problem (2023), directed by Ekaterina Eremenko. You can watch the trailer here.
The movie is based on the paper Compact Bonnet Pairs by Alexander I. Bobenko, Tim Hoffmann, and Andrew O. Sageman-Furnas.
Curvature lines of isothermic Torus
Let $f$ denote the parameterization function of the isothermic torus in the animations below. It has one family of planar curvature lines, i.e. the curve $u \mapsto f(u, v_{0})$ lies in a plane for every fixed $v_{0}$.
It also has another family of spherical curvature lines, i.e. the curve $v \mapsto f(u_{0}, v)$ lies on a sphere for every fixed $u_{0}$.
Darboux’s classification of isothermic surfaces with planar curvature lines
A parameter $w_0$ generates a family of curves, e.g. the following curve
which can then be used to generate a surface with planar curvature lines:
The Bonnet pairs
Below are animations of the Bonnet tori $f_{+}$ and $f_{-}$