Planar discrete isothermic nets of conical type

Abstract

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 Mobius transformations. Furthermore, we use the underlying circle pattern to characterize discrete isothermic nets in the projective model of Mobius 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.