Geometrically smooth spline bases for data fitting and simulation

Given a topological complex $M$ with glueing data along edges shared by adjacent faces, we study the associated space of geometrically smooth spline functions that satisfy differentiability properties across shared edges. We present new and efficient constructions of basis functions of the space of $G^{1}$-spline functions on quadrangular meshes, which are tensor product b-spline functions on each quadrangle and with b-spline transition maps across the shared edges. This new strategy for constructing basis functions is based on a local analysis of the edge functions, and does not depend on the global topology of $M$. We show that the separability of the space of $G^{1}$ splines across an edge allows to determine the dimension and a basis of the space of $G^{1}$ splines on $M$.This leads to explicit and effective constructions of basis functions attached to the vertices, edges and faces of $M$.This basis construction has important applications in geometric modeling and simulation. We illustrate it by the fitting of point clouds by $G^{1}$ splines on quadrangular meshes of complex topology and in Isogeometric Analysis methods for the solution of diffusion equations. The ingredients are detailed and experimentation results showing the behavior of the method are presented.