## Xianfeng David GuAssociate Professor Department of Computer Science Department of Applied Mathematics State University of New York at Stony Brook Room 419 New Computer Science Building State University of New York at Stony Brook Stony Brook, New York 11794-2424 Phone: (631) 632-1828 (Office) Fax: (631) 632-8334 gu at cs.stonybrook.edu http://www.cs.stonybrook.edu/~gu Center of Visual Computing |

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.

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.

**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.

**Genus zero closed surface**, the conformal mapping can be commputed usingSpherical Harmonic Maps .**Genus zero surface with multiple boundary components**, the conformal mapping can be computed usingGeneralized Koebe's Method , or slit map method.**Genus one surface**, the conformal mapping can be commputed using holomorphic one forms. Another algorithm is to use Euclidean Ricci flow.**Higher genus surfaces**, the mapping can be commputed usingHyperbolic Ricci Flow .**Quasiconformal mappings**can be computed usingAuxiliary Metric Method .

- Graphics: surface global conformal parameterization.
- Medical Imaging: conformal brain mapping and colon flattening.
- Geometric Modeling: manifold splines.
- Vision: 3D shape analysis, surface matching and recognition.
- Wireless sensor network: routing, load balancing.