Optimal Shape Preserving Properties Research Articles

An Optimal Property of B-Bases for the Modified Richardson Method

A space with a normalized totally positive basis has a unique normalized B-basis. In computer-aided geometric design, normalized B-bases present optimal shape-preserving properties. More optimal properties of normalized B-bases were proved previously. This paper provides a new optimal property concerning the modified Richardson iterative method when applied to collocation matrices of normalized B-bases. Moreover, a similar optimal property is proved for the tensor product of normalized B-bases.

A New Class of Trigonometric B-Spline Curves

We construct one-frequency trigonometric spline curves with a de Boor-like algorithm for evaluation and analyze their shape-preserving properties. The convergence to quadratic B-spline curves is also analyzed. A fundamental tool is the concept of the normalized B-basis, which has optimal shape-preserving properties and good symmetric properties.

Extremal and optimal properties of B-bases collocation matrices

Totally positive matrices are related with the shape preserving representations of a space of functions. The normalized B-basis of the space has optimal shape preserving properties. B-splines and rational Bernstein bases are examples of normalized B-bases. Some results on the optimal conditioning and on extremal properties of the minimal eigenvalue and singular value of the collocation matrices of normalized B-bases are proved. Numerical examples confirm the theoretical results and answer related questions.

A totally positive basis for circle approximations

We construct a six-dimensional space of polynomials containing approximations of trigonometric functions. In this way, an approximation of a circle can be represented as a parametric curve in this space. We show that the space has shape preserving representations and we construct the basis with optimal shape preserving properties.

Control point based exact description of curves and surfaces, in extended Chebyshev spaces

Extended Chebyshev spaces that also comprise the constants represent large families of functions that can be used in real-life modeling or engineering applications that also involve important (e.g. transcendental) integral or rational curves and surfaces. Concerning CAGD, the unique normalized B-bases of such vector spaces ensure optimal shape preserving properties, important evaluation or subdivision algorithms and useful shape parameters. Therefore, we propose global explicit formulas for the entries of those transformation matrices that map these normalized B-bases to the traditional (or ordinary) bases of the underlying vector spaces. Then, we also describe general and ready to use control point configurations for the exact representation of those traditional integral parametric curves and (hybrid) surfaces that are specified by coordinate functions given as (products of separable) linear combinations of ordinary basis functions. The obtained results are also extended to the control point and weight based exact description of the rational counterpart of these integral parametric curves and surfaces. The universal applicability of our methods is presented through polynomial, trigonometric, hyperbolic or mixed extended Chebyshev vector spaces.

On a class of shape-preserving refinable functions with dilation 3

We analyze the properties of a class of shape-preserving refinable functions with dilation M=3. We give an algorithm to construct totally positive bases with optimal shape-preserving properties on a finite interval. Bernstein-type bases on [0,1] are also treated. Moreover, semiorthogonal wavelets associated with these refinable functions are constructed. Finally, a detailed example is described.

Optimal bases for a class of mixed spaces and their associated spline spaces

We consider spaces of the form span 〈 1 , t , … , t n − 4 , u 1 ( t ) , u 2 ( t ) , u 3 ( t ) , u 4 ( t ) 〉 , where the functions u i ( i = 1 , … , 4 ) are algebraic polynomials, or trigonometric or hyperbolic functions. We find intervals [ 0 , α ] where we can guarantee that the spaces possess normalized totally positive bases (and so, shape preserving representations). We construct their normalized B -bases (with optimal shape preserving properties). We also present a unified approach to deal with the associated spline spaces. A recursive procedure to obtain bases with optimal shape preserving and stability properties is presented.

Progressive iterative approximation and bases with the fastest convergence rates

All normalized totally positive bases satisfy the progressive iterative approximation property. The normalized B-basis has optimal shape preserving properties and we prove that it satisfies the progressive iterative approximation property with the fastest convergence rates. A similar result for tensor product surfaces is also derived.

Shape preserving alternatives to the rational Bézier model

We discus several alternatives to the rational Bézier model, based on using curves generated by mixing polynomial and trigonometric functions, and expressing them in bases with optimal shape preserving properties (normalized B-bases). For this purpose we develop new tools for finding B-bases in general spaces. We also revisit the C-Bézier curves presented by Zhang (1996), which coincide with the helix spline segments developed by Pottmann and Wagner (1994), and are nothing else than curves expressed in the normalized B-basis of the space P 1= span{1,t, cost, sint} . Such curves provide a valuable alternative to the rational Bézier model, because they can deal with both free form curves and remarkable analytical shapes, including the circle, cycloid and helix. Finally, we explore extensions of the space P 1 , by mixing algebraic and trigonometric polynomials. In particular, we show that the spaces P 2= span{1,t, cost, sint, cos2t, sin2t} , Q= span{1,t,t 2, cost, sint} and I= span{1,t, cost, sint,t cost,t sint} are also suitable for shape preserving design, and we find their normalized B-basis.

Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

In a recent article, Ge et al. (1997) identify a special class of rational curves (Harmonic Rational Bézier (HRB) curves) that can be reparameterized in sinusoidal form. Here we show how this family of curves strongly relates to the class of p-Bézier curves, curves easily expressible as single-valued in polar coordinates. Although both subsets do not coincide, the reparameterization needed in both cases is exactly the same, and the weights of a HRB curve are those corresponding to the representation of a circular arc as a p-Bézier curve. We also prove that a HRB curve can be written as a combination of its control points and certain Bernstein-like trigonometric basis functions. These functions form a normalized totally positive B-basis (that is, the basis with optimal shape preserving properties) of the space of trigonometric polynomials {1, sin t, cos t, …. sin mt, cos mt} defined on an interval of length < π.

Shape preserving representations for trigonometric polynomial curves

This paper has two main goals. Firstly, we show that the space of trigonometric polynomials T m = span{1, cos t, sin t, ..., cos mt, sin mt} is not suitable for those methods of CAGD which use control polygons. It is well-known that the bases with good shape preserving properties are the normalized totally positive bases and we prove here that T m does not possess normalized totally positive bases. Secondly, we show that the space C m = span{1, cos t, ..., cos mt} is suitable for design purposes using control polygons. In fact, we construct a basis C m of C m with optimal shape preserving properties and analyze some aspects for the computation of the corresponding curves.

Totally positive bases for shape preserving curve design and optimality of B-splines

Normalized totally positive (NTP) bases present good shape preserving properties when they are used in Computer Aided Geometric Design. Here we characterize all the NTP bases of a space and obtain a test to know if they exist. Furthermore, we construct the NTP basis with optimal shape preserving properties in the sense of (Goodman and Said, 1991), that is, the shape of the control polygon of a curve with respect to the optimal basis resembles with the highest fidelity the shape of the curve among all the control polygons of the same curve corresponding to NTP bases. In particular, this is the case of the B-spline basis in the space of polynomial splines. Further examples are given.

Shape preserving representations and optimality of the Bernstein basis

This paper gives an affirmative answer to a conjecture given in [10]: the Bernstein basis has optimal shape preserving properties among all normalized totally positive bases for the space of polynomials of degree less than or equal ton over a compact interval. There is also a simple test to recognize normalized totally positive bases (which have good shape preserving properties), and the corresponding corner cutting algorithm to generate the Bezier polygon is also included. Among other properties, it is also proved that the Wronskian matrix of a totally positive basis on an interval [a, ∞) is also totally positive.