Abstract. We consider discrete quasi-interpolants based on C 1 quadratic boxsplines on uniform criss-cross triangulations of a rectangular domain. The main problem consists in finding good (if not best) coefficient functionals, associated with boundary box-splines, giving both an optimal approximation order and a small infinity norm of the operator. Moreover, we want that these functionals only involve data points inside the domain. They are obtained either by minimizing their infinity norm w.r.t. a finite number of free parameters, or by inducing superconvergence of the operator at some specific points lying near or on the boundary.
Given a non-uniform criss-cross triangulation of a rectangular domain Ω , we consider the approximation of a function f and its partial derivatives, by general C 1 quadratic spline quasi-interpolants and their derivatives. We give error bounds in terms of the smoothness of f and the characteristics of the triangulation. Then, the preceding theoretical results are compared with similar results in the literature. Finally, several examples are proposed for illustrating various applications of the quasiinterpolants studied in the paper.
The aim of this paper is to investigate, in a bounded domain of R 3 , two blending sums of univariate and bivariate C 1 quadratic spline quasi-interpolants.The main problem consists in constructing the coefficient functionals associated with boundary generators, i.e. generators with supports not entirely inside the domain. In their definition, these functionals involve data points lying inside or on the boundary of the domain. Moreover, the weights of these functionals must be chosen so that the quasi-interpolants have the best approximation order and a reasonable infinite norm.We give their explicit constructions, infinite norms and error estimates. In order to illustrate the approximation properties of the proposed quasiinterpolants, some numerical examples are presented and compared with those obtained by some other trivariate quasi-interpolants given recently in the literature.
This paper investigates the space S (X) generated by the integer translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions that forms a regular partition of the space into tetrahedra.In S (X) local spline quasi-interpolants are defined by linear combinations of integer translates of B {B α , α ∈ Z 3 } and local linear functionals {D f (α), α ∈ Z 3 }. First a quasi-interpolant of differential type is proposed and its coefficient functionals are linear combinations of values of f with its partial derivatives at the center of the support of B α . Then, by convenient discretisations of the above differential coefficients, three quasi-interpolants of discrete type with different properties are defined. In this case the coefficient functionals are linear combinations of values of f at specific points in the support of B α .Upper bounds both for the norm of the discrete quasi-interpolation operators and for the approximation error are given.Finally, some numerical examples, illustrating the approximation properties of the proposed quasi-interpolants, are presented.
Please cite this article in press as: Dagnino, C., et al. Curve network interpolation by C 1 quadratic B-spline surfaces. Comput. Aided Geom. Des. (2015), http://dx.
Highlights• Interpolation of a B-spline curve network by a surface based on C 1 quadratic B-splines on criss-cross triangulations.• Proof of the existence and uniqueness of the surface and constructive algorithm for its generation.• Numerical and graphical results and comparisons with other spline methods.
AbstractIn this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate C 1 quadratic splines on criss-cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods.
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