We study the rigidity of polyhedral surfaces using variational principles. The action
functionals are derived from the cosine laws. The main focus of this paper is on the
cosine law for a nontriangular region bounded by three possibly disjoint geodesics.
Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By
studying the derivative of the cosine laws, we discover a uniform approach to several
variational principles on polyhedral surfaces with or without boundary. As a
consequence, the work of Penner, Bobenko and Springborn and Thurston on rigidity
of polyhedral surfaces and circle patterns are extended to a very general
context.
Keywords
derivative cosine law, energy function, variational
principle, edge invariant, circle packing metric, circle
pattern metric, polyhedral surface, rigidity, metric,
curvature
The Center of Mathematical
Science
Zhejiang University
Hangzhou, Zhejiang 310027, China
and
Department of Mathematics
Rutgers University
Piscataway, NJ 08854
USA