A checkerboard pattern approach to isothermic surfaces
Felix Dellinger  1, 2@  
1 : Vienna University of Technology
2 : Technische Universität Graz

A checkerboard pattern is a quadrilateral net with Z2 combinatorics where every second face is a parallelo-
gram. 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 discrete principal curvature nets is invariant under Möbius transformations and
can be studied in the projective model of Möbius geometry. Koenigs nets are exactly those nets that allow
a discrete dualization. Analogously to the smooth case, they can be characterized by the existence of certain
osculating conics or by the equality of their Laplace invariants.
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 and their Goursat transformations.

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