Jesús M. Carnicer

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 Bézier 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.

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

Abstract 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.

Total positivity and optimal bases

In a finite dimensional space which has a totally positive basis there exist special bases, called B-bases, such that they generate all totally positive bases by means of totally positive matrices. B-bases are optimal totally positive bases in several senses. From the geometrical point of view, B-bases correspond to the bases with optimal shape preserving properties. From a numerical point of view B-bases are least supported and least conditioned bases among all totally positive bases of space. In order to deal with these questions we introduce a partial order in the set of nonnegative bases: if (v 0,..., v n ) = (u 0,..., u n )H for a nonnegative matrix H, we say that (u 0,..., u n ) ≤ (v 0,..., v n ) . We shall show that B-bases are minimal for this partial order.

Convexity of rational curves and total positivity

We analyze convexity preserving properties of curves from a geometric point of view. We also characterize totally positive systems of functions in terms of geometric convexity preserving properties of the rational curves. Rational Bezier and nonuniform rational B-spline curves are included in this setting.

Interpolation on lattices generated by cubic pencils

Principal lattices are distributions of points in the plane obtained from a triangle by drawing equidistant parallel lines to the sides and taking the intersection points as nodes. Interpolation on principal lattices leads to particularly simple formulae. These sets were generalized by Lee and Phillips considering three-pencil lattices, generated by three linear pencils. Inspired by the addition of points on cubic curves and using duality, we introduce an addition of lines as a way of constructing lattices generated by cubic pencils. They include three-pencil lattices and then principal lattices. Interpolation on lattices generated by cubic pencils has the same good properties and simple formulae as on principal lattices.

Convexity preserving interpolation and Powell-Sabin elements

Abstract This paper is concerned with constructing local convexity preserving bivariate interpolants. Piecewise linear upper and lower bounds for such interpolants are derived which induce certain partitions of the underlying domain. The fact that these partitions are data dependent and cannot be constructed by a local procedure reflects the difficulty in constructing local convexity preserving methods. However, simple additional conditions are established which ensure that Powell-Sabin elements provide C 1 convexity preserving interpolants.

Generation of lattices of points for bivariate interpolation

Abstract Principal lattices in the plane are distributions of points particularly simple to use Lagrange, Newton or Aitken–Neville interpolation formulae. Principal lattices were generalized by Lee and Phillips, introducing three-pencil lattices, that is, points which are the intersection of three lines, each one belonging to a different pencil. In this contribution, a semicubical parabola is used to construct lattices of points with similar properties. For the construction of new lattices we use cubic pencils of lines and an addition of lines on them.

Classification of bivariate configurations with simple Lagrange interpolation formulae

In 1977 Chung and Yao introduced a geometric characterization in multivariate interpolation in order to identify distributions of points such that the Lagrange functions are products of real polynomials of first degree. We discuss and describe completely all these configurations up to degree 4 in the bivariate case. The number of lines containing more nodes than the degree is used for classifying these configurations.

Piecewise linear interpolants to Lagrange and Hermite convex scattered data

This paper concerns two fundamental interpolants to convex bivariate scattered data. The first,u, is the supremum over all convex Lagrange interpolants and is piecewise linear on a triangulation. The other,l, is the infimum over all convex Hermite interpolants and is piecewise linear on a tessellation. We discuss the existence, uniqueness, and numerical computation ofu andl and the associated triangulation and tessellation. We also describe how to generate convex Hermite data from convex Lagrange data.

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.