Paolo Costantini

On a class of weak Tchebycheff systems

AbstractIn this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formn n requiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.

Curve and surface construction using variable degree polynomial splines

Abstract The aim of this paper is to describe applications of variable degree polynomials in the area of curve and surface construction. These polynomials have the same simple structure and the same properties as cubics with the advantage of a strong control on their shape, given by two degrees which play the role of design parameters. As a consequence, more flexible C2 B-spline or NURBS like curves and C2 tensor-product or Boolean sum surfaces can be obtained with the same geometric construction and the same computational cost of their cubic counterparts.

Computation of optimal composite re-parameterizations

Rational re-parameterizations of a polynomial curve that preserve the curve degree and [0,1] parameter domain are characterized by a single degree of freedom. The optimal re-parameterization in this family (that comes closest under the L2 norm to arc-length parameterization) can be identified by solving a quadratic equation, but may exhibit too much residual parametric speed variation for motion control and other applications. Closer approximations to arc-length parameterizations require more flexible re-parameterization functions, such as piecewise-polynomial/rational forms. We show that, for fixed nodes, the optimal piecewise-rational parameterization of the same degree is defined by a simple recursion relation, and we analyze its convergence to the arc-length parameterization. With respect to the new curve parameter, this representation is only of C^0 continuity, although the smoothness and geometry of the curve are unchanged. A C^1 parameterization can be obtained by using continuity conditions, rather than optimization, to fix certain free parameters, but the objective function is then highly non-linear and does not admit a closed-form optimization. Empirical results from implementations of these methods are presented.

Piecewise monotone quadratic histosplines

A method is presented for constructing a quadratic spline function satisfying area-matching conditions and local monotonicity constraints, according to the frequencies on the class intervals and to the shape of a given histogram. Such a function is “as close as possible” to the quadratic spline that satisfies the area-matching conditions and the minimum curvature property and generally exhibits a visually pleasing graph.

Shape-Preserving Approximation of Spatial Data

We present a new method for the construction of shape-preserving curves approximating a given set of 3D data, based on the space of “quintic like” polynomial splines with variable degrees recently introduced in [7]. These splines – which are C3 and therefore curvature and torsion continuous – possess a very simple geometric structure, which permits to easily handle the shape-constraints.

Shape-Preserving C 3 Interpolation: The Curve Case.

We present a new method for the construction of shape-preserving curves interpolating a given set of 3D data. The interpolating functions are obtained using “quintic-like” spaces of polynomial splines with variable degrees. These splines are of class C3 and are therefore curvature and torsion continuous and possess a very simple geometric structure, which permits to easily handle the shape-constraints.

Shape-Preserving Approximation by Space Curves

This paper describes a new method for the construction of C2 shape-preserving curves which approximate an ordered set of data in R3. The curves are obtained using the variable degree polynomial spline spaces recently described in [5].

Shape-preserving interpolation with variable degree polynomial splines

The main goal of this paper is to present some results obtained in functional shape-preserving interpolation using variable degree polynomial splines, and show how these functions are emerging as a powerful tool both in tension methods and in CAGD applications.

A local shape-preserving interpolation scheme for scattered data

Abstract In this paper we present a new local method for constructing a C 1 function which interpolates triangulated scattered data sets. This function is made up by polynomial triangular macro-elements of adaptive degree, which are an extension of the Clough–Tocher cubic elements and tend to the interpolating planar triangular interpolant as the degrees tend to infinity.

On discrete hyperbolic tension splines

A hyperbolic tension spline is defined as the solution of a differential multipoint boundary value problem. A discrete hyperbolic tension spline is obtained using the difference analogues of differential operators; its computation does not require exponential functions, even if its continuous extension is still a spline of hyperbolic type. We consider the basic computational aspects and show the main features of this approach.