**Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization**

*Bobenko, Alexander I. ; Sechelmann, Stefan ; Springborn, Boris* (2016)

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrila...

**Discrete complex analysis on planar Quad-graphs**

*Bobenko, Alexander I. ; Günther, Felix* (2016)

We develop further a linear theory of discrete complex analysis on general quad-graphs, extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph leads to generalizations as well as to new proofs of previously known discrete analogs of classical theorems. New results include in particular di...

**S-conical CMC surfaces. Towards a unified zheory of discrete surfaces with constant mean curvature**

*Bobenko, Alexander I. ; Hoffmann, Tim* (2016)

We introduce a novel class of s-conical nets and, in particular, study s-conical nets with constant mean curvature. Moreover we give a unified description of nets of various types: circular, conical and s-isothermic. The later turn out to be interpolating between the circular net discretization and the s-conical one.