  Spline functions have long been used in numerical solution of differential
equations. Recently it revives as isogeometric analysis, which offers
integration of finite element analysis and NURBS based CAD into a single
unified process. Usually many NURBS pieces are needed to build geometrically
continuous CAD models. In this paper, we introduce some multiply periodic
splines defined on hyperbolic disc. A single piece of such splines is enough to
build complex CAD models. Multiresolution analysis on surfaces of high genus
built from such splines can be carried out naturally. CAD and FEA are
integrated directly on such models. It is difficult to derive such splines,
only a theoretical framework is presented, together with some simple examples.
Rigorous derivation and construction of B-splines will be given in future
papers.
