The Projective Linear Transition Map for Constructing Smooth Surfaces

We exhibit the essentially unique projective linear (rational linear) reparameterization for constructing C^s surfaces of genus g>0. Conversely, for quadrilaterals and isolated vertices of valence 8, we show constructively for s=1,2 that this map yields a projective linear spline space for surfaces of genus greater or equal to 1. This establishes the reparametrization to be the simplest possible transition map.