The theory of circle packing has roots in complex analysis, hyperbolic geometry and topology of three manifolds.
On a triangulated closed surface, a circle packing is a configuration of circles on the surface such that each circle corresponds to a vertex and a pair of adjacent circles corresponds to an edge. Fix a relationship between each pair of adjacent circles, we are interested in the existence and uniqueness of a circle packing.
Variational principles are applied to study such problems. The energy functions of the variational principles are related to the volume of some hyperbolic polyhedra. The rigidity of circle packing follows from the concavity of the energy functions.
The theory of circle packing has applications in discrete differential geometry, discrete conformal geometry, approximation of Riemann mappings, computational conformal geometry. It supplies new algorithms to computer graphics and medical visualization.