discrete differential geometry
My research lies in the area of discrete differential geometry. You can find an introduction in my lecture notes I gave at Kyushu University, Japan in 2008.
recent preprints
- Tim Hoffmann and Gudrun Szewieczek. Isothermic nets with spherical parameter lines from discrete holomorphic maps, 2024. [ bib | arXiv ]
- Jannik Steinmeier Tim Hoffmann, Andrew O. Sageman-Furnas. Skew parallelogram nets and universal factorization, 2024. [ bib | arXiv ]
mathematical visualization
![](gallery/iso60.png)
An isothermic torus with one family of planar and one family of spherical curvature lines that generates a compact Bonnet pair (Bonnet pairs are non congruent pairs of surfaces that are isometric and have the same mean curvature). We could prove the existence of Bonnet tori here.
![](gallery/erfRings.png)
An Erf ring pattern. Orthogonal Ring patterns are a generalization of orthogonal circle patterns: Each ring consists of a pair of concentric circles and for each lattice direction one of them intersects its corresponding neighbour orthogonally. The theory behind them is written here.
![](gallery/dogLanternRender.png)
Discrete orthogonal geodesic nets (DOG nets for short) are a discretization of developable surfaces, that utilizes the fact that a surface is developpable if and only if it admits an orthogonal net of geodesics. The advantage of such a parametrization by orthogonal geodesics — compared to one by rulings for example — is, that it is invariant under isometric deformations. One can read about them here and here.
![](gallery/assoc.png)
A surface in the associated family of an s-isothermic Enneper surface. S-isothermic nets are made from orthogonal circle patterns (although the associated family of them is no longer orthogonal). One can read about them here.
A 3d printed circle pattern on the sphere, that stereographically projects to the regular pattern in the plane. This can be seen as the shadow of the circles under a point light at the northpole is exactly the stereographic projection.
![](gallery/BTclock.png)
![](images/text10.png)