Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines
Introduction
Pythagorean hodograph planar curves (for short, PH curves) form a remarkable family of parametric polynomial curves with many useful features and characteristics. They have been extensively studied during the last three decades, see (Farouki, 2008), (Pottmann, 1995; Farouki et al., 1998; Pelosi et al., 2007; Ait-Haddou et al., 2008; Jüttler, 2001; Sir and Jüttler, 2007; Choi et al., 2008; Bastl et al., 2013; Kosinka and Lávička, 2014; Albrecht et al., 2017; Fang and Wang, 2018; Farouki, 2019; Farouki et al., 2019), and other references therein. In particular, it is well known that, among all cubic planar parametric curves, cubic PH curves can be characterised by geometric properties of their control polygons: equal interior angles and constant ratios between consecutive segments (Farouki and Sakkalis, 1990). This geometric characterisation naturally establishes a one-to-one correspondence between the class of all cubic PH curves with obtuse interior angles and Hermite interpolation problems (two points and associated tangent directions) with appropriate configuration of the data (tight data). Repeated application of this one-to-one correspondence naturally produces a unique PH-spline curve interpolating any given sequence of such Hermite tight data (Ait-Haddou and Biard, 1994; Meek and Walton, 1997). However, if the data are associated with a given knot-vector, the unique corresponding cubic PH spline function cannot be , save for exceptional configurations.
The impossibility of obtaining a cubic PH spline function interpolating Hermite data can be viewed as a manifestation of the lack of flexibility of polynomial spaces, due to the fact that no parameter is inherently attached to them. This is the reason why, in many situations, it can be useful to replace them by their most natural generalisations, that is, by Extended Chebyshev spaces (Karlin and Studden, 1966; Schumaker, 1981; Lyche, 1985; Pottmann, 1993; Mazure, 1999a, Mazure, 2004). Though more difficult to handle, such spaces present the great advantage to inherently possess parameters which can be used to improve the unique solutions to given problems. Probably the most famous example is provided by the so-called tension splines introduced in (Schweikert, 1966) to eliminate undesired oscillations in cubic spline interpolation. All pieces were taken from the cubic-like Extended Chebyshev space spanned by the functions , where a is any positive parameter whose well-known global effect is to produce “ piecewise affine” interpolants at +∞.
In the present paper, the cubic space will be replaced by any space spanned by four functions of the form , , where ℓ is a positive integer, which is an Extended Chebyshev space on any interval contained in . The integer ℓ represents the number of missing monomials with respect to the degree polynomial space, and for this reason we call it the sparsity of this cubic-like space which itself is said to be sparse. Sparse spaces are especially interesting for design purposes, because all design algorithms with these spaces are hardly more complicated than with cubic spaces (Mazure, 1999b; Ait-Haddou et al., 2013a). Moreover, they inherently possess two parameters, first the sparsity parameter ℓ, second the interval parameter related to where they operate, which proved to produce powerful shape effects for spline design (Laurent et al., 1997; Mazure, 2001). Recently, sparse spaces were used to construct cubic-like PH curves (Ait-Haddou and Mazure, 2017), with the advantage of some flexibility resulting from the presence of their two parameters. The most important result to retain from (Ait-Haddou and Mazure, 2017) is the characterisation of cubic-like sparse PH curves by geometric properties of their control polygons, which extends the one concerning their cubic counterparts. Along with the presence of parameters, this characterisation will enable us to obtain cubic-like sparse PH splines based on a fixed knot-vector, interpolating associated Hermite tight data. This is the object of the present work.
The necessary background on cubic-like sparse spaces and associated splines is briefly presented in Section 2. In particular, we recall why it is recommended that sparse splines be defined after disconnecting the interval parameters from the knot-vector through a positive piecewise affine function. This is crucial for geometric design, for it simultaneously permits to take full advantage of the parameters offered by sparse spaces and overcome their lack of symmetry. The crucial geometric characterisation of cubic-like sparse PH curves by means of their control polygons, and the flexibility they permit, are summarised in Section 3. These results are applied to Hermite interpolation of tight data by cubic-like sparse PH spline curves in Section 4. As a matter of fact, this question had already been addressed in (Ait-Haddou and Mazure, 2017). However, here, being concerned with continuity, we are not only interested in the resulting curves, but in cubic-like sparse PH spline functions, based on a given knot-vector associated with the tight data to be interpolated. For this reason, inspired by what is recommended for design, we first have to revisit the definition of cubic-like sparse PH curves. The Hermite interpolation problem can now be solved via infinitely many different spline functions preserving the possible symmetry properties of the given tight data. Among this infinitely many solutions, infinitely many are . How to construct the PH segments of such solutions, one after the other, is explained in Section 5. This progressive method is then illustrated with several examples of tight data taken from classical curves, with special insistence on symmetry preservation. Through one example we also illustrate what can be done when the data are not tight, according to the pre-processing step suggested in (Jaklič et al., 2010). Our results are synthesized and commented in Section 6 with a view to possible future work.
Section snippets
Design with sparse cubic-like Müntz spaces and splines
Given any numbers , the -dimensional space spanned by the functions , , is called a Müntz space. The space is an Extended Chebyshev space (for short, EC-space) on , in the sense that any non-zero element of this space vanishes at most n times on . Suppose that . Then, the (n-dimensional) space obtained through the ordinary differentiation D is in turn an EC-space on , and the EC-space is said to be good for design on .
Cubic-like sparse PH curves
In this section we need to briefly explain how to recognise that a given cubic-like sparse curve is a PH curve, and how to construct such PH curves. The results summarised below were obtained in (Ait-Haddou and Mazure, 2017).
Interpolation of Hermite data with sparse cubic-like PH spline curves: a revisit
By cubic-like sparse PH-spline curves we mean spline curves composed of cubic-like sparse PH segments, the sparsity being allowed to depend on the segment and even to be zero at some places. By comparison, by cubic PH-spline curves we mean that all segments are standard cubic PH curves. In this section we revisit the Hermite interpolation by cubic-like sparse PH-spline curves already addressed in (Ait-Haddou and Mazure, 2017).
Hermite interpolation by cubic-like PH-spline curves
The data are the same as in Theorem 4. For the sake of simplicity, from now on we assume that the knots are , . Among the infinite number of solutions to the Hermite interpolation problem (17) – resulting from the parameters inherent in cubic-like sparse spaces (as re-defined in Section 4.2) – can we find one which is on ? We can give an affirmative answer to this question, thus solving the problem
Final comments
Replacing cubic spaces by the larger framework of cubic-like sparse spaces has enabled us to construct PH spline functions interpolating given tight Hermite data, and based on given knot-vectors. This is due to the free parameters involved in cubic-like sparse spaces, which even permit to obtain infinitely many solutions to such interpolation problems. The method presented in this article is progressive: we construct the cubic-like sparse PH spline function piece after piece, in order
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The second author gratefully acknowledges support from INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico, Italy.
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