Xianfeng David Gu


Associate Professor
Department of Computer Science
State University of New York at Stony Brook

Room 2425 Computer Science Building
State University of New York at Stony Brook
Stony Brook, New York 11794-4400

Phone:  (631) 632-1828 (Office) 
Fax: (631) 632-8334
gu at cs.sunysb.edu

http://www.cs.sunysb.edu/~gu
Center of Visual Computing

Computational Conformal Geometry

Conformal Structure is a natural geometric structure on surfaces, which governs many physics phenomena, such as heat diffusion, electric-magnetic fields, etc. Conformal field theory plays fundamental role in string theory.

In mathematics, conformal means "angle preserving". Conformal structure is a specail atlas of the surface, such that angles among tangent vectors can be coherently defined on different local coordinate systems. Furthermore, concepts in complex anylasis can be defined on the surface via conformal structure. Conformal geometry is the intersection of algebraic geometry, differential geometry, complex anylasis and algebraic topology.

In engineering, conformal structure is between topological structure and geometric structure, which is more rigid than topology and more flexible than geometry. Therefore, conformal structure leads to canonical non-rigid deformation, which is important for engineering applications, especially for shape anylasis, classification and registration.

The goal of computational conformal geometry is to convert concepts and theorems from Riemann surface theory to practical algorithms, and implement them for engineering applications.

Riemann Surface and Holomorphic Differentials

A surface with a conformal structure is called a Riemann surface. All metric surfaces are Riemann surfaces. The image illustrates the conformal structure using isothermal coordinates. The algorithm is based on computing holomorphic differentials on Riemann surfaces.

Uniformization and Ricci Flow

Surface uniformization means that all metric surfaces can be conformally mapped to one of the three canonical domains, the sphere, the plane and the hyperbolic space. The figures show the uniformization for surfaces with various topologies. Unformization converts general 3D geometric problems to 2D problems in these canonical domains. Ricci flow is a powerful geometric analytic tool, which has been applied to prove Poincare conjecture. Ricci flow is a parabolic system of partial differential equations which acts like the heat equation to spread the curvature of a Riemannian metric evenly over the surface to produce a metric of constant curvature. Computational discrete Ricci flow is the practical method to compute surface uniformization, it has many important applications in many engineering fields.

Algorithms

Algorithms for computing conformal mappings from a surface onto the canonical domain can be summarized as follows:
  1. Genus zero closed surface, the conformal mapping can be commputed using Spherical Harmonic Maps.
  2. Genus zero surface with multiple boundary components, the conformal mapping can be computed using Generalized Koebe's Method, or slit map method.
  3. Genus one surface, the conformal mapping can be commputed using holomorphic one forms. Another algorithm is to use Euclidean Ricci flow.
  4. Higher genus surfaces, the mapping can be commputed using Hyperbolic Ricci Flow.
  5. Quasiconformal mappings can be computed using Auxiliary Metric Method.

Applications

Applications of computational conformal geometry in engineering fields are innumerous. It offers the rigorous theoretic frame work for computing surface matching, registration, classification and analysis. The followings are a few examples in the most directly related fields: