B-spline-like bases for C2C^2 cubics on the Powell-Sabin 12-split

For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a B\'ezier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for $C^0$-, $C^1$-, and $C^2$-smoothness are derived

Bivariate Hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in $\mathbb{R}^3$ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points $\{(x_i, y_i)\}_{i=1}^N$ from which none of the pairs $(x_i,y_i)$ and $(x_j,y_j)$ with $i\neq j$ coincide, it is proved that the resulting surface (function) is $C^1$. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is $C^2$ if the data are not 'too irregular'. Finally the approximation properties of the methods are investigated

A Hermite interpolatory subdivision scheme for C2C^2-quintics on the Powell-Sabin 12-split

In order to construct a $C^1$-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme. In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally $C^3$ and globally $C^2$. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.Comment: 17 pages, 7 figure

Recent Results on Near-Best Spline Quasi-Interpolants

Roughly speaking, a near-best (abbr. NB) quasi-interpolant (abbr. QI) is an approximation operator of the form $Q_af=\sum_{\alpha\in A} \Lambda_\alpha (f) B_\alpha$ where the $B_\alpha$'s are B-splines and the $\Lambda_\alpha (f)$'s are linear discrete or integral forms acting on the given function $f$. These forms depend on a finite number of coefficients which are the components of vectors $a_\alpha$ for $\alpha\in A$. The index $a$ refers to this sequence of vectors. In order that $Q_a p=p$ for all polynomials $p$ belonging to some subspace included in the space of splines generated by the $B_\alpha$'s, each vector $a_\alpha$ must lie in an affine subspace $V_\alpha$, i.e. satisfy some linear constraints. However there remain some degrees of freedom which are used to minimize $\Vert a_\alpha \Vert_1$ for each $\alpha\in A$. It is easy to prove that $\max \{\Vert a_\alpha \Vert_1 ; \alpha\in A\}$ is an upper bound of $\Vert Q_a \Vert_{\infty}$: thus, instead of minimizing the infinite norm of $Q_a$, which is a difficult problem, we minimize an upper bound of this norm, which is much easier to do. Moreover, the latter problem has always at least one solution, which is associated with a NB QI. In the first part of the paper, we give a survey on NB univariate or bivariate spline QIs defined on uniform or non-uniform partitions and already studied by the author and coworkers. In the second part, we give some new results, mainly on univariate and bivariate integral QIs on {\sl non-uniform} partitions: in that case, NB QIs are more difficult to characterize and the optimal properties strongly depend on the geometry of the partition. Therefore we have restricted our study to QIs having interesting shape properties and/or infinite norms uniformly bounded independently of the partition

Quasi-Interpolation in a Space of C 2 Sextic Splines over Powell–Sabin Triangulations

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell– Sabin triangulations. These spline functions are of class C 2 on the whole domain but fourth-order regularity is required at vertices and C 3 regularity is imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell–Sabin triangles with a small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden’s identity from a more explicit version of the control polynomials introduced some years ago in the literature. Finally, some tests show the good performance of these operators.Erasmus+ International Dimension programme, European CommissionPAIDI programme, Junta de Andalucía, Spai

On C2 cubic quasi-interpolating splines and their computation by subdivision via blossoming

We discuss the construction of C2 cubic spline quasi-interpolation schemes defined on a refined partition. These schemes are reduced in terms of degrees of freedom compared to those existing in the literature. Namely, we provide a rule for reducing them by imposing super-smoothing conditions while preserving full smoothness and cubic precision. In addition, we provide subdivision rules by means of blossoming. The derived rules are designed to express the B-spline coefficients associated with a finer partition from those associated with the former one."Maria de Maeztu" Excellence Unit IMAG (University of Granada, Spain) CEX2020-001105-MICIN/AEI/10.13039/501100011033University of Granada University of Granada/CBU

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods

On numerical quadrature for C1C^1 quadratic Powell-Sabin 6-split macro-triangles

The quadrature rule of Hammer and Stroud [16] for cubic polynomials has been shown to be exact for a larger space of functions, namely the $C^1$ cubic Clough-Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle [21]. We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of $C^1$ quadratic Powell-Sabin 6-split macro-triangles. We show that the $3$-node Gaussian quadrature(s) for quadratics can be generalised to the $C^1$ quadratic Powell-Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in [21]. The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three. For the inner split-point at the barycentre, we prove that the two 3-node quadratic polynomial quadratures of Hammer and Stroud exactly integrate also the $C^1$ quadratic Powell-Sabin spline space if and only if the edge split-points are at their respective edge midpoints. For other positions of the inner and edge split-points we provide numerical examples showing that three nodes suffice to integrate the space exactly, but a full classification and a closed-form solution in the generic case remain elusive

Interpolation and scattered data fitting on manifolds using projected Powell–Sabin splines

We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(Uξ , ξ)}ξ∈ satisfying certain conditions of smooth dependence on ξ. If is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function