• Differential geometry and discrete differential geometry
• Mathematical visualization / Computer graphics
• Industrial/architectual geometry
• Multitouch and tangible interfaces
• Software development

Discrete Minimal Surfaces of Koebe Type
Alexander I. Bobenko, Ulrike Bücking, and Stefan Sechelmann, in Najman L., Romon P. (eds) Modern Approaches to Discrete Curvature. Lecture Notes in Mathematics, vol 2184. Springer, Cham, 2017.
There is also an electronic version of the article that contains rotatable 3D models of the images: 3D Version

Variational Methods for Discrete Surface Parameterization. Applications and Implementation. (Doctoral Thesis)
Stefan Sechelmann. Variational Methods for Discrete Surface Parameterization. Applications and Implementation, TU-Berlin, 2016, DOI: 10.14279/depositonce-5415

Discrete conformal maps: Boundary value problems, circle domains, Fuchsian and Schottky uniformization
Alexander I. Bobenko, Stefan Sechelmann, and Boris Springborn. Discrete conformal maps: Boundary value problems, circle domains, Fuchsian and Schottky uniformization. In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.

DGD Gallery: Storage, sharing, and publication of digital research data
M. Joswig, M. Mehner, S. Sechelmann, J. Techter, and A.I. Bobenko. DGD Gallery: Storage, sharing, and publication of digital research data. In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.

S-conical minimal surfaces. Towards a unified theory of discrete minimal surfaces.
Alexander I. Bobenko, Benno König, Tim Homann, and Stefan Sechelmann. S-conical minimal surfaces. Towards a unified theory of discrete minimal surfaces. preprint: http://www.discretization.de

Discrete Uniformization of Polyhedral Surfaces with Non-positive Curvature and Branched Covers over the Sphere via Hyper-ideal Circle Patterns
Alexander Bobenko, Nikolay Dimitrov, and Stefan Sechelmann.
Discrete & Computational Geometry, March 2017, Volume 57, Issue 2, pp 431–469
Preprint: Discrete uniformization of finite branched covers over the Riemann sphere via hyper-ideal circle patterns (arXiv:1510.04053)

Surface panelization using periodic conformal maps
Thilo Rörig, Stefan Sechelmann, Agata Kycia, and Moritz Fleischmann, Surface panelization using periodic conformal maps, Advances in Architectural Geometry 2014 (P. Block, J. Knippers, N.J. Mitra, and W. Wang, eds.), Springer, 2014, p. 365. Best Paper Award AAG 2014 (Conference report)

Uniformization of discrete Riemann surfaces
Stefan Sechelmann, Uniformization of discrete Riemann surfaces, Discrete Differential Geometry (Alexander I. Bobenko, Richard Kenyon, Peter Schröder, and Günter M. Ziegler, eds.), vol. 9, Oberwolfach Reports, no. 3, European Mathematical Society, 2012, pp. 2105-2107.

Quasiisothermic Mesh Layout
S. Sechelmann, T. Rörig, and A. I. Bobenko. Quasiisothermic Mesh Layout. In Hesselgren, L.; Sharma, S.; Wallner, J.; Baldassini, N.; Bompas, P.; Raynaud, J. (Eds.). Advances in Architectural Geometry 2012. 2012, 344 p. 285 illus. in color. ISBN 978-3-7091-1250-2

Topology Optimisation of Regular and Irregular Elastic Gridshells by means of a Non-linear Variational Method.
E. Lafuente E, S. Sechelmann, T. Rörig, and C. Gengnagel. Topology Optimisation of Regular and Irregular Elastic Gridshells by means of a Non-linear Variational Method. In Hesselgren, L.; Sharma, S.; Wallner, J.; Baldassini, N.; Bompas, P.; Raynaud, J. (Eds.). Advances in Architectural Geometry 2012. 2012, 344 p. 285 illus. in color. ISBN 978-3-7091-1250-2

On the Materiality and Structural Behaviour of highly-elastic Gridshell Structures
E. Lafuente E, C. Gengnagel, S. Sechelmann, and T. Rörig. On the Materiality and Structural Behaviour of highly-elastic Gridshell Structures In Gengnagel, C.; Kilian, A.; Palz, N.; Scheurer, F. (Eds.). Computational Design Modeling: Proceedings of the Design Modeling Symposium Berlin 2011 2012, XVI, 347 p. ISBN 978-3-642-23435-4

Discrete Minimal Surfaces, Koebe Polyhedra, and Alexandrov's Theorem. Variational Principles, Algorithms, and Implementation. (Diploma Thesis)

• Variational Methods for Discrete Surface Parameterization. Applications and Implementation, Thesis Defense Berlin April 25. 2016
• Uniformization of Elliptic and Hyperelliptic Curves via Discrete Conformal Equivalence, Oberwolfach 2016
• VaryLab - Discrete Surface Optimization, Copenhagen 2015
• Periodic conformal maps, Berlin 2014
• Constructing discrete minimal surfaces of s-conical type from orthogonal circle patterns: examples, Schney 2013
• Uniformization of discrete Riemann surfaces, Obergurgl 2012
• Quiasiisothermic Mesh Layout, AAG Paris 2012
• Glas, Stahl und Geometrie - Mathematik in der modernen Architektur, Urania Berlin 2012
• Halfedge!, Berlin 2011
• U3D - 3D content to mathematical papers, Berlin 2008
• jRWorkspace Plugin Java SDK, Berlin 2008

Riemann surfaces have been studied for a long time. Recently, the development of robust conformal mapping algorithms based on triangulaions has lead to significant interest in discrete versions of smooth theorems and constructions in the context of Riemann surfaces. In this section I present a selection of images from my research on the topic of discrete Riemann surfaces.

A Riemann surface can have quite different representations. I consider three kinds of representations in my work: Embeddings in 3-space, algebraic curves, and quotient spaces of the Euclidean plane, the hyperbolic plane, and the Riemann sphere. The latter beeing uniformizations of Fuchsian type or schottky type respectively.

Most of the algorithms for the calculation with discrete Riemann surfaces are available online as part of the VaryLab project.

Fuchsian uniformization of the hyperelliptic curve $\mu^2=\lambda\cdot(\lambda^6-1)$.
Riemann map of a slit square to the disk with circular boundary components.
Uniformization of algebraic curve of Lawsons surface $\mu^2=\lambda^6-1$.
Algebraic curve of Lawsons surface. Boundary and identifications of a fundamental domain.
Schottky data for a genus 3 discrete Riemann surface.
Fuchsian uniformization of genus 3 discrete Riemann surface given by Schottky data.
Symmetric uniformization of Lawsons surface $\mu^2=\lambda^6-1$.
Euclidean uniformization of a genus 2 surface with translational periods.
Map to the corresponding algebraic curve with 6 branch points. Mercator projection.

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