Computational conformal geometry is an emerging interdisciplinary field, combining modern geometry theories from pure mathematics with computational algorithms from computer science and offering rigorous and practical tools for tackling massive geometric data processing problems. From a theoretical point of view, computational conformal geometry has deep roots in mathematics and physics. In mathematics, it is at the intersection of many fields, such as algebraic topology, differential geometry, Riemann surface, harmonic analysis, and so on. It also has close relations to many fields of physics, such as the electromagnet field in electrodynamics, elasticity deformation in mechanics, heat diffusion in thermal dynamics, modular space theory in super string theory. From a practical point of view, computational conformal geometry offers many powerful tools to handle a broad range of geometric problems in the engineering world.
With the great efforts of both mathematicians and computer scientists, major breakthroughs in conformal geometry have been made in recent years. Several theoretical frameworks in the field of conformal geometry have been systematically developed. Many computational conformal algorithms have been invented and applied to engineering and medical fields. There are, however, still many profound facts in conformal geometry. The discretization method and the computational strategy are still widely open. Furthermore, the surge of practical applications has advanced the computational algorithms of this field, but the theoretical foundations need to be rigorously laid down in the near future.