FactsInstitute of Mathematics@TU-Berlin
SFB / Transregio 109 Discretization in Geometry and Dynamics
Project A01: Discrete Riemann Surfaces
Blog: Notes on things
office: MA 879
phone: +49(0)30 314-29486
Publications(click on the image to download the full-text PDF)
Variational Methods for Discrete Surface Parameterization. Applications and Implementation. (Doctoral Thesis)
Discrete conformal maps: Boundary value problems, circle domains, Fuchsian and Schottky uniformization
DGD Gallery: Storage, sharing, and publication of digital research data
S-conical minimal surfaces. Towards a unified theory of discrete minimal surfaces.
Discrete uniformization of finite branched covers over the Riemann sphere via hyper-ideal circle patterns
Surface panelization using periodic conformal maps
Uniformization of discrete Riemann surfaces
Quasiisothermic Mesh Layout
Topology Optimisation of Regular and Irregular Elastic Gridshells by means of a Non-linear Variational Method.
On the Materiality and Structural Behaviour of highly-elastic Gridshell Structures
TalksA list of talks in chronological order.
Discrete Riemann Surfaces
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.
VaryLabVaryLab is all about mesh optimization, we say discrete surface optimization. That means you can modify a given mesh to have minimal energy in a certain sense. The energy in question is a combination of energies that are defined on the vertex positions of the input mesh. VaryLab implements various energies for discrete surfaces, e.g., planarity of faces, equal lengths of edges, curvature of parameter curves and many more.
Visit us at http://www.varylab.com
I work together with the people of www.interactive-scape.com to create multi-touch applications.
jPETScTaojPETScTao is the name for a Java project, that tries to make a part of the functionality of two numeric libraries PETSc and Tao accessible for Java programs. It utilizes the JNI to achieve that.
Discrete S-Isothermic Minimal Surfaces
Schoen's I-6 Surface
Discrete Minimal Surfaces This application calculates discrete minimal surfaces as described in my diploma thesis. The construction method follows the approach of Bobenko, Hoffmann, and Springborn. It uses the notion of discrete isothermic surfaces and their Cristoffel transform to define discrete minimal surfaces. The program is able to create surfaces with planar boundary curvature lines.
Alexandrov's Polyhedron In cooperation with Ivan Izmestiev I implemented an algorithm for constructing convex polyhedra with a given metric. The java webstart on the left is the main program which I used for testing and research purposes.
Teamgeist(TM) Polyhedron The right program is an application of the alexandrov program. It calculates a polyhedron with predefined combinatorics and symetry of the Teamgeist(TM) soccer ball. I created this during the world soccer championchips. For a description and a java applet see A "Teamgeist" Polyhedron
Another polyhedron I created with the help of this tool is the Reuleaux Triangle Tetrahedron. This is a Tetrahedron with curved sides which are Reuleaux Triangles. Those triangles get slighly bent to fit together.
Koebe Polyhedron Editor Together with Boris Springborn I created a visualization tool for Koebe Polyhedra. Click the image to start a Java Webstart application.
The left section is a graph designer. If the graph is 3-connected and enbedded the program calculates the corresponding polyhedron and displays a normalized representation in the righ section.
This is the first program I created for the geometry group. It contains numerical algorithms for nonlinear optimization of the convex functional involved. I use the MTJ library for linear solving and sparse matrix representation. The program is also part of my diploma thesis and is being described in detail in Chapter 1.
Piano PlayingHere I publish the piano recordings I'm doing for fun and at home. Feel free to download and listen. For recording I use a Seiler 132 Konzert and the software sample Ivory.
Last modified: Stefan Sechelmann 2016-07-28