In this work, we use some results concerning the connection between blossoms and splines, especially those related to the smoothness conditions to develop an algorithm for constructing on an ...arbitrary triangulation a C2 spline approximant with minimal degree. Numerical tests are presented to illustrate the theoretical results.
An extension to triangular domains of the univariate q-Bernstein basis functions is introduced and analysed. Some recurrence relations and properties such as partition of unity and degree elevation ...are proved for them. It is also proved that they form a basis for the space of polynomials of total degree less than or equal to n on a triangle. In addition, it is presented a de Casteljau type evaluation algorithm whose steps are all linear convex combinations.
•The referees’ comments have been incorporated into the new version of the manuscript, correcting many typos.•The structure of the paper has been improved and the number of figures has been ...reduced.•Some maintained graphs have been combined to reduce the number of figures.•Some tables have been combined for the same purpose.•The notation used for the quasi-interpolation operators has been improved.•In the quadratic case, a very interesting comment by reviewer 3 has been incorporated, which relates the quasi-interpolation scheme introduced in the article to a quasi-interpolant obtained by discretisation of the linear functional of the classical quadratic differential quasi-interpolant.•The constructive method introduced in the article has the potential to give rise to new quasi-interpolants that incorporate specific properties.
In this paper we propose the construction of univariate low-degree quasi-interpolating splines in the Bernstein basis, considering C1 and C2 smoothness, specific polynomial reproduction properties and different sets of evaluation points. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values. Moreover, we get quasi-interpolating splines with special properties, imposing particular requirements in case of free parameters. Finally, we provide numerical tests showing the performances of the proposed methods.
This work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions (splines), which uses the blossoming approach. Some existing results in the ...literature are reformulated, such as the smoothness conditions between polynomials of a spline, by using the affinity property of the blossom. Some definitions of sub-splines are proposed which can be very useful in the study and the construction of splines such as macro-elements or quasi-interpolants. As an application of the proposed results, a C1 quartic spline quasi-interpolant with optimal approximation order is defined without using any mask for smoothness or B-spline basis. Numerical results are presented and compared with other methods given in the literature.
The paper deals with the characterization of Powell-Sabin triangulations allowing the construction of bivariate quartic splines of class C2. The result is established by relating the triangle and ...edge split points provided by the refinement of each triangle. For a triangulation fulfilling the characterization obtained, a normalized representation of the splines in the C2 space is given.
The construction of the generalized Bézier model with shape parameters is one of the research hotspots in geometric modeling and CAGD. In this paper, a novel shape-adjustable generalized Bézier (or ...SG-Bézier, for short) surface of order (m, n) is introduced for the purpose to construct local and global shape controllable free-form complex surfaces. Meanwhile, some properties of SG-Bézier surfaces and the influence rules of shape parameters, as well as the constructions of special triangular and biangular SG-Bézier surfaces, are investigated. Furthermore, based on the terminal properties and linear independence of SG-Bernstein basis functions, the conditions for G1 and G2 continuity between two adjacent SG-Bézier surfaces are derived, and then simplified them by choosing appropriate shape parameters. Finally, the specific steps and applications of the smooth continuity for SG-Bézier surfaces are discussed. Modeling examples show that our methods in this paper are not only effective and can be performed easily, but also provide an alternative strategy for the construction of complex surfaces in engineering design.
We have derived functions of the lowest possible degree that enable us to evaluate curvature monotonicity for any 2D and 3D rational Bézier curves. We proved that the degree of the function is at ...most 8n−12 for planar rational Bézier curves of degree n, and is at most 11n−18 for space rational Bézier curves of degree n. These functions are derived in the Bernstein basis, allowing for efficient checking of curvature monotonicity using subdivision or Bézier clipping. As an application, we present real-time visualization of the region of a particular control point that guarantees monotonic variation of curvature over the entire segment of the rational Bézier curve. This allows users to identify where to move the control point to ensure that the curvature changes monotonically.
•Curvature Monotonicity Evaluation Functions (CMEFs) are derived for Bézier curves•These CMEFs cover polynomial/rational planar/space Bézier curves of any degree•Derived CMEFs have succinct structures in Bernstein basis•Exact degree reduction mechanism in the derivation process is clarified•CMEFs can be used to interactive curve design with monotonically varying curvature
The aim of this paper is to study generating functions for the coefficients of the classical superoscillatory function associated with weak measurements. We also establish some new relations between ...the superoscillatory coefficients and many well-known families of special polynomials, numbers, and functions such as Bernstein basis functions, the Hermite polynomials, the Stirling numbers of second kind, and also the confluent hypergeometric functions. Moreover, by using generating functions, we are able to develop a recurrence relation and a derivative formula for the superoscillatory coefficients.
This paper is concerned with a new generalization of rational Bernstein–Bézier curves involving q-integers as shape parameters. A one parameter family of rational Bernstein–Bézier curves, weighted ...Lupaş q–Bézier curves, is constructed based on a set of Lupaş q-analogue of Bernstein functions which is proved to be a normalized totally positive basis. The generalized rational Bézier curve is investigated from a geometric point of view. The investigation provides the geometric meaning of the weights and the representation for conic sections. We also obtain degree evaluation and de Casteljau algorithms by means of homogeneous coordinates. Numerical examples show that weighted Lupaş q–Bézier curves have more modeling flexibility than classical rational Bernstein–Bézier curves and Lupaş q–Bézier curves, and meanwhile they provide better approximations to the control polygon than rational Phillips q–Bézier curves.
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