Solving Bonnet problems to construct families of surfaces

In this work, the authors study Bonnet Problems using Cartan moving frames and associated structure equations. The Cartan structural forms are written in terms of the first and second fundamental forms, and the Lax system is consequently reinterpreted; orthonormal moving frames are obtained solutions to this Bonnet-Lax system, via numerical integration. Certain classifications of families of surfaces are provided in terms of the first and second fundamental forms, given certain prescribed invariants. Numerical applications (an improved Runge Kutta method) applied to this theoretical framework produced a fast way to visualize families of surfaces under investigation. We provide a few examples and visual models.