Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology

We analyze the space of geometrically continuous piecewise polynomial functions, or splines, for rectangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G1G1 spline functions, we introduce the concept of topological surface with gluing data attached to the edges shared by faces. The framework does not require manifold constructions and is general enough to allow non-orientable surfaces. We describe compatibility conditions on the transition maps so that the space of differentiable functions is ample and show that these conditions are necessary and sufficient to construct ample spline spaces. We determine the dimension of the space of G1G1 spline functions which are of degree ⩽k   on triangular pieces and of bi-degree ⩽(k,k)⩽(k,k) on rectangular pieces, for k   big enough. A separability property on the edges is involved to obtain the dimension formula. An explicit construction of basis functions attached respectively to vertices, edges and faces is proposed; examples of bases of G1G1 splines of small degree for topological surfaces with boundary and without boundary are detailed.