## FG Geometrie und Visualisierung

9 Items

**On a discretization of confocal quadrics. I. An integrable systems approach**

*Bobenko, Alexander I. ; Schief, Wolfgang K. ; Suris, Yuri B. ; Techter, Jan* (2016-08-04)

Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a conformal parametrization along curvature lines, or, equivalently, supporting or...

**Discrete confocal quadrics and checkerboard incircular nets**

*Techter, Jan* (2021)

Confocal quadrics constitute a special example of orthogonal coordinate systems. In this cumulative thesis we propose two approaches to the discretization of confocal coordinates, and study the closely related checkerboard incircular nets. First, we propose a discretization based on factorizable solutions to an integrable discretization of the Euler-Poisson-Darboux equation. The constructed so...

**Structures in three-dimensional Euclidean space from hyperbolic tilings**

*Kolbe, Benedikt Maximilian* (2020)

In this thesis we introduce a theory of isotopy classes of tilings with given symmetry group on hyperbolic surfaces, possibly punctured, nonorientable, and with boundary. We first generalise combinatorial tiling theory to incorporate tilings with tiles that are not closed disks. To establish a combinatorial theory for isotopy classes of tilings, based on Delaney-Dress tiling theory, we subseque...

**Isothermic constrained Willmore tori**

*Tervooren, Jonas* (2020)

This dissertation treats the classification of isothermic constrained Willmore tori. Jörg Richter proved in his dissertation that for every immersion f : M → R³ of an isothermic constrained Willmore torus, there exists a conformal change of the euclidean metric of R³ such that the surface has constant mean curvature (CMC) in a space form. We extended his proof such a way that it remains true if...

**Polyhedral surfaces of constant curvature and discrete uniformization**

*Kourimská, Hana* (2020)

In this thesis we introduce a new discretization of the Gaussian curvature on piecewise flat surfaces. It is defined on the conical singularities of the surface and it is the quotient of the angle defect and the area of the Voronoi cell. We investigate the existence and uniqueness of metrics with constant discrete Gaussian curvature within discrete conformal classes of piecewise flat surfaces. ...

**Discrete Yamabe problem for polyhedral surfaces**

*Kourimska, Hana ; Springborn, Boris* (2019-09-13)

We introduce a new discretization of the Gaussian curvature on piecewise at surfaces. As the prime new feature the curvature is scaled by the factor 1/r2 upon scaling the metric globally with the factor r. We develop a variational principle to tackle the corresponding discrete uniformisation theorem – we show that each piecewise at surface is discrete conformally equivalent to one with constant...

**Discrete Yamabe problem for polyhedral surfaces**

*Kourimska, Hana ; Springborn, Boris* (2019-09-13)

We introduce a new discretization of the Gaussian curvature on piecewise at surfaces. As the prime new feature the curvature is scaled by the factor 1/r2 upon scaling the metric globally with the factor r. We develop a variational principle to tackle the corresponding discrete uniformisation theorem – we show that each piecewise at surface is discrete conformally equivalent to one with constant...

**Conformal maps from a 2-torus to the 4-sphere**

*Bohle, Christoph ; Leschke, Katrin ; Pedit, Franz ; Pinkall, Ulrich* (2012)

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle o...

**Isothermic submanifolds of symmetric R-spaces**

*Burstall, Francis E. ; Donaldson, Neil M. ; Pedit, Franz ; Pinkall, Ulrich* (2011)

We extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christo¤el, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric R-spaces with essentially no loss of integrable structure.