Abstract
We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes. A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.
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Acknowledgments
The first author would like to thank Alexander Bobenko and Thomas Banchoff for their inspiring lectures.
Compliance with ethical standards The authors declare that they have no conflict of interest and this article does not contain any studies with human participants or animals performed by any of the authors.
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Communicated by Ngaiming Mok.
This research was supported by the DFG Collaborative Research Centre SFB/TRR 109 Discretization in Geometry and Dynamics. The first author was partially supported by Berlin Mathematical School and the Croucher Foundation of Hong Kong.
