 Fakultät II
Institut für Mathematik

# Virtual Math Labs: Polyhedral Surfaces Geometry Group

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Contents:

To start a lab just click on the screenshot. If the application does not start, have a look at our Help page. ### Cyclidic nets

Explore 2D and 3D cyclidic nets and associated geometric structures
(Implemented by Emanuel Huhnen-Venedey)

Cyclidic nets are a discretization of orthogonal nets, closely related to the discretization as circular nets. In Euclidean 3-space they discretize curvature line parametrized surfaces resp. triply orthogonal coordinate systems.

The application allows to extend (fixed) 2- and 3-dimensional circular nets to cyclidic nets, where the extension is uniquely determined by the choice of an orthonormal 3-frame at one vertex. The user may change this frame interactively and explore the corresponding change of the cyclidic net. The program also allows to examine single cyclidic patches and whole Dupin cyclides.

#### Literature

A. I. Bobenko, E. Huhnen-Venedey. Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with Dupin cyclides Geometriae Dedicata Vol. 159, Issue 1 (2012), 207-237 (online). ### Minimal Surface Editor

Construction of discrete minimal surfaces with boundary conditions
(Implemented by Stefan Sechelmann)

Construct discrete minimal surfaces from the combinatorics of their curvature lines. A detailed description is contained in the Diploma Thesis of Stefan Sechelmann.

#### Literature

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 Diploma Thesis unpublished (online).

A. I. Bobenko, T. Hoffmann, and B. A. Springborn. Minimal surfaces from circle patterns: Geometry from combinatorics Annals of Mathematics 164 (2006), 231-264 (online).

A. I. Bobenko, B. A. Springborn. Variational principles for circle patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356 (2004), 659-689 (online). ### Alexandrov Polyhedron Editor

Construction of a convex polytope from a given development
(Implemented by Stefan Sechelmann)

For any convex polyhedral metric on the sphere there exists a unique convex polytope that has this metric on its boundary.

Usage: In the left half of the window draw a triangulation and assign lengths to the edges. Start the calculation. If your data defines a convex polyhedral metric, then in the right half of the window you will see the convex polytope.

#### Literature

A. I. Bobenko, I. Izmestiev. Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes. Preprint arXiv:math.DG/0609447, 2006. ### Koebe Polyhedron Editor

Construction of the Koebe polyhedron for a given 3-connected planar graph
(Implemented by Stefan Sechelmann)

For each combinatorial type of convex 3-dimensional polyhedra, there exists a unique representative with the following properties:
1. All edges are tangent to the unit sphere.
2. The barycenter of the points where the edges touch the sphere is the origin.
This application constructs this Koebe polyhedron for a given 3-connected planar graph.

#### Literature

A. I. Bobenko, B. A. Springborn. Variational principles for circle patterns and Koebe's theorem. Trans. Amer. Math. Soc. 356 (2004), 659-689 (online).

B. A. Springborn. A unique representation of polyhedral types. Centering via Möbius transformations. Math. Z. 249:3 (2005), 513-517 (online).

B. A. Springborn. Variational principles for circle patterns. PhD thesis, Technische Universität Berlin, 27 Nov 2003. Supervisor: A. I. Bobenko. Published online. Errata page [PDF]. ### Top Visualization

Program for the 3D visualization of a rigid body motion around a fixed point in a homogeneous gravity field at the example of quader. It includes the construction of the special cases of Kowalewski and Goryachev-Chaplygin, displays their integrals of motion and visualizes their corresponding elastic rods. Further it includes the possibility to compare different discretizations of the paths of lagrange tops, i. e. the integrable discretization developed in  and a numerical one.

For a complete description of content, usage and functionality see .
Besides the java web start there is a zip archiv of a stand alone version (Windows or Linux) of the program

#### Literature

 A. I. Bobenko and Yu- B. Suris, A Discrete Time Lagrange Top and Discrete Elastic Curves, Amer. Math. Soc. Transl. (2) Vol. 201, 2000.

 René Bodack, Dynamics of the Spinning Top: Discretization and Interactive Visualization, Diploma Thesis, unpublished (online).

 Boris Springborn . 12.10.2011.