Planar discrete isothermic nets of conical type

  • Christian Müller
Original Paper


We explore a specific discretization of isothermic nets in the plane which can also be interpreted as a discrete holomorphic map. The discrete orthogonality of the quadrilateral net is achieved by the so called conical condition imposed on vertex stars. That is, the sums of opposite angles between edges around all vertices are equal. This conical condition makes it possible to define a family of underlying circle patterns for which we can show invariance under Möbius transformations. Furthermore, we use the underlying circle pattern to characterize discrete isothermic nets in the projective model of Möbius geometry, and as Moutard nets in homogeneous coordinates in relation to the light cone in this model. We further investigate some examples of discrete isothermic nets and apply the Christoffel dual construction to obtain discrete minimal surfaces of conical type.


Discrete differential geometry Conical nets Isothermic nets Minimal surfaces 

Mathematics Subject Classification

51B10 52B70 52C26 53A10 



This research was supported in part by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ through Grant I 706-N26 of the Austrian Science Fund (FWF).


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Copyright information

© The Managing Editors 2015

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyViennaAustria

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