Conformal Geometry in Engineering and Medicine

With the development of 3D geometric acquisition technologies, massive 3D geometric data are ubiquitous today. It is of great challenges to process and analyze 3D geometric data efficiently and accurately. Computational conformal geometry is an emerging inter-disciplinary field, which combines modern geometry with computer science and offers rigorous and practical tools for tackling massive geometric data processing problems. The concepts and methods in conformal geometry play fundamental roles in many fields in engineering and medicine.

Conformal geometry studies the invariants under the conformal transformation (angle preserving mapping) group. Conformal geometry is more flexible than Riemannian geometry, and more rigid than topology. Conformal geometry is capable of unifying all shapes in real world to one of three canonical shapes, the sphere, the plane, or the hyperbolic disk; conformal geometric algorithms convert 3D geometric processing problems to 2D image processing problems; furthermore, all surfaces in real life have conformal structures, therefore conformal geometric methods are general. These merits make conformal geometry a powerful tool for real applications.