Esmeralda Mainar

Shape preserving alternatives to the rational Bézier model

Abstract We discus several alternatives to the rational Bezier 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-Bezier 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, cos t, sin t} . Such curves provide a valuable alternative to the rational Bezier 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, cos t, sin t, cos 2t, sin 2t} , Q= span {1,t,t 2 , cos t, sin t} and I = span {1,t, cos t, sin t,t cos t,t sin t} are also suitable for shape preserving design, and we find their normalized B-basis.

A general class of Bernstein-like bases

This paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials. Bases with optimal shape preserving and stability properties are constructed. Evaluation and subdivision algorithms are provided. Bases for the corresponding and mixed spline spaces are also constructed. Nice properties of these bases and the generated curves are shown.

Error analysis of corner cutting algorithms

Corner cutting algorithms are used in different fields and, in particular, play a relevant role in Computer Aided Geometric Design. Evaluation algorithms such as the de Casteljau algorithm for polynomials and the de Boor–Cox algorithm for B‐splines are examples of corner cutting algorithms. Here backward and forward error analysis of corner cutting algorithms are performed. The running error is also analyzed and as a consequence the general algorithm is modified to include the computation of an error bound.

A basis of C-Bézier splines with optimal properties

It is proved that the C-B-spline segments introduced by Zhang are generated by a normalized totally positive basis. It is also constructed a normalized B-basis of C-B-splines, which presents optimal shape preserving and stability properties.

Corner cutting algorithms associated with optimal shape preserving representations

Given a space of functions which admits shape preserving representations using control polygons, we construct a corner cutting algorithm which will be called B-algorithm. It is an evaluation algorithm satisfying important properties such as subdivision property and convergence to the curve of the resulting control polygons. Many examples are given.

Shape preservation regions for six-dimensional spaces

Abstract We analyze the critical length for design purposes of six-dimensional spaces invariant under translations and reflections containing the functions 1, cos t and sin t. These spaces also contain the first degree polynomials as well as trigonometric and/or hyperbolic functions. We identify the spaces whose critical length for design purposes is greater than 2π and find its maximum 4π. By a change of variables, two biparametric families of spaces arise. We call shape preservation region to the set of admissible parameters in order that the space has shape preserving representations for curves. We describe the shape preserving regions for both families.

Quadratic-cycloidal curves

Interesting curves can be represented in the space Q=span{1,t,t2,cos t,sin t}, t∈[0,α] (0<α<2π). In this paper we introduce quadratic-cycloidal B-splines associated to equally spaced knots and the properties of the generated curves for 0<α<π. It is proved that, when α→0, the limit of a QC-B-spline curve approaches a B-spline curve of degree 4.

Representing circles with five control points

We show that five is the minimal dimension of a space required to draw a complete circle with a unique control polygon. We identify all five-dimensional spaces invariant under translations and reflections where we can find shape preserving representations of a circle parameterized by its arc length.

Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials

It has been recently pointed out that several orthogonal polynomials of the Askey table admit asymptotic expansions in terms of Hermite and Laguerre polynomials [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]. From those expansions, several known and new limits between polynomials of the Askey table were obtained in [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]. In this paper, we make an exhaustive analysis of the three lower levels of the Askey scheme which completes the asymptotic analysis performed in [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]: (i) We obtain asymptotic expansions of Charlier, Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Hermite polynomials. (ii) We obtain asymptotic expansions of Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Charlier polynomials. (iii) We give new proofs for the known limits between polynomials of these three levels and derive new limits.

Monotonicity preserving representations of non-polynomial surfaces

We prove that, in contrast to the case for rational surfaces, some tensor product representations through spaces containing algebraic, trigonometric and hyperbolic polynomials are monotonicity preserving. The surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product Bezier and tensor product B-spline surfaces.