Lecture Notes for Computational Conformal Geometry 2010

In the summer of 2010, the Mathematics Science Center of Tsinghua university invited me to give a short course on computational conformal geometry. The students were majored in mathematics, computer science or other engineering, medical science fields from Tsinghua university, Peking university and other universities.

The course focused on the fundamental theories and computational algorithms for conformal geometry. The profound theorems, such as Gauss-Bonnet, Hodge decomposition, surface Ricci flow, Uniformization were proven in the discrete setting. The major concepts, such as homology/cohomology, fundamental group, holomorphic differential forms, constant curvature metrics, canonical conformal representations, conformal modules, were calculated using computational algorithms, and visualized by modern graphics techniques. The course aims at using computational approach and visualization techniques to teach abstract geometric theories.

The summer program is sponsored by the Mathematics Science Center of Tsinghua University. I am very grateful to the great support from the Mathematics Science Center during the course.

The followings are the slides for the lectures. The textbook based on the lectures will be published in near future. The lecture notes are under constant improvement and revision. Please forward your comments and criticisms to gu@cs.sunysb.edy.

Lecture Slides

1. Canonical surface conformal mapping [pdf]

This lecture introduces the canonical conformal mappings for compact metric surfaces with boundaries.

2. Planar conformal mapping, Escher-Droste effect [pdf]

conformal mappings between flat tori are explained in details. Their lifts to the covering spaces have been applied to image processing and visual effects.

3. Conformal structure [pdf]

This lecture explains the concepts of conformal strucutre, Riemann surface, Teichmuller space.

4. Mobius transformation [pdf]

This lecture introduces the concept of Mobius transformation group, their classification, and properties. Especially, Mobius transformation preserves cross ratios, thereofore preserves hyperbolic metric.

5. Discrete surface [pdf]

Discrete surfaces, discrete Riemannian metrics, discrete Gaussian curvature, discrete Laplace-Beltrami operator.

6. Simplicial homology [pdf]

7. Fundamental group [pdf]

fundamental group, covering space, deck transformation,

8. Topological algorithm [pdf]

Topological algorithms, homology group, cohomology group, CW-cell decomposition, fundamental group.

9. Half edge data structure [pdf]

Data structure for discrete surfaces.

10. Characteristic class [pdf]

Topological obstruction, Poincare-Hopf theorem, unit tangent bundle

11.Combinatorial Hodge decomposition [pdf]

Combinatorial Hodge decomposition theorem.

12. de Rham cohomology [pdf]

Exterior calculus, de Rham cohomology.

13. Holomorphic 1-form [pdf]

Conformal mapping algorithms based on holomorphic 1-forms.

14. Harmonic mapping [pdf]

Spherical conformal mapping algorithm based on non-linear heat diffusion method.

15. Heat diffusion [pdf]

16. Movable frame [pdf]

Movable frame method, Gauss Egregium theorem,

17. Euclidean Ricci flow [pdf]

Euclidean Ricci flow

18. Inversive distance Ricci flow, Yamabe flow[pdf]

Euclidean inversive distance Ricci flow, Yamabe flow.

19. Hyperbolic Ricci flow [pdf]

Hyperbolic Ricci flow.

20. Graph drawing [pdf]

Graph drawing.