Geometry Workshop Obergurgl 2021

A checkerboard pattern is a quadrilateral net with Z2 combinatorics where every second face is a parallelogram. Such a mesh can be easily obtained through midpoint subdivision from a general quadrilateral net. The structure of a checkerboard pattern is very suitable to describe discrete differential geometric properties of the net. In particular we can use it to consistently define conjugate nets, principal curvature nets, a shape operator and Koenigs nets. We find that the class of principal curvature nets is invariant under Möbius transformations and Koenigs nets are exactly those nets that allow a discrete dualization. Isothermic nets can then be characterized as Koenigs nets that are also principal curvature nets. Again we can transform them using the Möbius transformation or dualization. The combination of both allows us to easily create examples of discrete minimal surfaces.