We propose a new paradigm for scattered data fitting with adaptive spline constructions based on the key interplay between parameterization and adaptivity. Specifically, we introduce two novel adaptive fitting schemes that combine moving parameterizations with adaptive spline refinement for highly accurate CAD models reconstruction from real-world scattered point clouds. The first scheme alternates surface fitting and data parameter optimization. The second scheme jointly optimizes the parameters and the surface control points. To combine the proposed fitting methods with adaptive spline constructions, we present a key treatment of boundary points. Industrial examples show that updating the parameterization, within an adaptive spline approximation framework, significantly reduces the number of degrees of freedom needed for a certain accuracy, especially if spline adaptivity is driven by suitably graded hierarchical meshes. The numerical experiments employ THB-splines, thus exploiting the existing CAD integration within the considered industrial setting, nevertheless, any adaptive spline construction can be chosen.

The theoretical basis of Floater's parameterization technique for triangulated surfaces is simultaneously a generalization (to non-barycentric weights) and a specialization (to a plane near-triangulation, which is an embedding of a planar graph with the property that all bounded faces are – possibly curved – triangles) of Tutte's Spring Embedding Theorem. Extensions of this technique cover surfaces with holes and periodic surfaces. The proofs presented previously need advanced concepts, such as rather involved results from graph theory or the theory of discrete 1-forms and consistent perturbations, or are not directly applicable to the above-mentioned extensions. We present a particularly simple geometric derivation of Tutte's theorem for plane near-triangulations and various extensions thereof, using solely the Euler formula for planar graphs. In particular, we include the case of meshes possessing a cylindrical topology – which has not yet been addressed explicitly but possesses important applications to periodic spline surface fitting – and we correct a minor inaccuracy in a previous result concerning Floater-type parameterizations for genus-1 meshes.

The coefficients of a tensor-product spline surface in R d with m × n control points form a tensor of order 3 and dimension ( m , n , d ) . Motivated by applications in isogeometric analysis we analyze the rank of this tensor. In particular, we propose a new construction for low rank tensor-product spline surfaces from given boundary curves. While the results of this construction are generally not affinely invariant, we propose a simple standardization procedure that guarantees affine invariance for d = 2 . In addition we provide a detailed comparison with existing constructions of spline surfaces from boundary data.