  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 splines defined on
branched covering of sphere, torus or general domains of $R^2$, which are
called branched splines in this paper. A single piece of such splines is enough
to build some 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. A theoretical framework is presented in
this paper, together with some simple examples.
