On the approximation order of a space data-dependent PH quintic Hermite interpolation scheme

doi:10.1016/j.cagd.2012.07.004

Computer Aided Geometric Design

Volume 30, Issue 1, January 2013, Pages 148-158

https://doi.org/10.1016/j.cagd.2012.07.004 Get rights and content

Abstract

Dealing with Pythagorean Hodograph quintic Hermite interpolation in the space, we deepen the analysis of the so-called CC criterion proposed in Farouki et al. (2008) for fixing the two free angular parameters characterizing the set of possible solutions, which remarkably influence the shape of the chosen interpolant. Such criterion is easy to implement, guarantees the reproduction of the standard cubic Hermite interpolant when it is a PH curve and usually allows the selection of interpolants with good shape. Here we first rigorously prove that the PH interpolant it selects doesnʼt depend on the unit pure vector chosen for representing its hodograph in quaternion form. Then we evaluate the corresponding interpolation scheme from a theoretical point of view, proving with the help of symbolic computation that it has fourth approximation order. A selection of experiments related to the spline implementation of the method confirms our analysis.

Highlights

► The approximation order of the CC data-dependent criterion for PH quintic space Hermite interpolation is studied, proving that it is four for smooth curves. ► The independence of the selection criterion on the unit vector used for the quaternion representation of the PH hodograph is also proved. ► A $C^{1}$ spline extension of the scheme is considered for the experiments.

References (10)

R.T. Farouki et al.
Structural invariance of spatial Pythagorean hodographs
Comput. Aided Geom. Design
(2002)
R.T. Farouki et al.
Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures
Comput. Aided Geom. Design
(2008)
H.I. Choi et al.
Clifford algebra, spin representation, and rational parameterization of curves and surfaces
Adv. Comp. Math.
(2002)
R.T. Farouki
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
(2008)
R.T. Farouki et al.
Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves
Adv. Comp. Math.
(2002)

There are more references available in the full text version of this article.

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Spatial quintic Pythagorean-hodograph interpolants to first-order Hermite data and Frenet frames
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It is notable that all of them produce the same cubic PH interpolant when the data are compatible to its existence. Later, Sestini et al. (2013) proved that the so-called “CC criterion” in that paper achieves the fourth-order approximation. Recently, Han et al. (2020) renewed the investigation of optimal interpolants in the quaternion space and proposed a new selection scheme such that the quaternion coefficient of the hodograph should form certain “extremal” positions.
We present a scheme to find spatial quintic Pythagorean-hodograph (PH) curves that interpolate given first-order Hermite data and Frenet frames. The two free parameters that appear in general quintic PH interpolants are determined to adjust the orientation of binormal vectors. Using the unique cubic interpolant to the given set of Hermite data as a reference, we produce a quintic PH interpolant that shares not only the first-order Hermite data but also the Frenet frames at the endpoints with the cubic counterpart. This approach can be readily applied to a sequence of points to generate a quintic PH $C^{1}$ spline curve with Frenet-frame continuity. We also prove that for any nonlinear analytic curve, such PH Frenet interpolants uniquely exist if the Hermite data are extracted from sufficiently close points on the curve.
G1 motion interpolation using cubic PH biarcs with prescribed length
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In this paper the $G^{1}$ interpolation scheme for motion data, i.e., interpolation of data points and rotations at the points, with cubic PH biarcs is presented. The rotational part of the motion is determined by the Euler–Rodrigues frame which matches the given boundary positions. In addition, the length of the biarc is prescribed. It is shown that the interpolant exists for any data and any chosen length greater than the difference between the interpolation points. The interpolant is given in a closed form and depends on some free shape parameters, which are determined so that the curve is of a nice shape and the twist of the Euler–Rodrigues frame is minimized. The spline construction is provided and numerical examples that confirm the derived theoretical results are included.
Curvature continuous path planning and path finding based on PH splines with tension
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Planar curvature continuous path generation with obstacle avoidance is considered by dealing with environments described either in terms of a structure directly specifying the boundaries of the obstacles (path planning) or by an adjacency matrix associated with uniform cells covering the whole scene (path finding). The second representation can be obtained for example from an image segmentation method when only an image of the scene is available. The method is composed by two main steps: the first is devoted to produce an intermediate admissible polyline, while the second defines a final curvature continuous path. Since the shape of the intermediate piecewise linear path remarkably influences the smoothness of the final path, different approaches are proposed to solve the path planning/finding problem in the first step. Then the vertices of the previously produced polyline are interpolated by using a $C^{1} \cap G^{2}$ interpolation scheme with tension based on Pythagorean–hodograph (PH) splines which have attractive features for applications. A strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. The results of several numerical experiments confirm the effectiveness of the method.
Planar C1 Hermite interpolation with PH cuts of degree (1, 3) of Laurent series
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Citation Excerpt :
There has been a great deal of study of PH curves by many researchers, not only of their formal representation (Albrecht and Farouki, 1996; Choi et al., 2001; Farouki, 1994) but also of practical aspects (Albrecht and Farouki, 1996; Choi et al., 2001; Farouki, 1994; Farouki and Neff, 1995; Jüttler, 2001; Moon, 1999). In particular, there has been a lot of work on the use of PH curves for interpolating several types of data-sets; planar data-sets (Albrecht and Farouki, 1996; Farouki and Neff, 1995; Jüttler, 2001; Kim et al., 2007; Walton and Meek, 1997) and spatial data-sets (Farouki et al., 2002; Pelosi et al., 2005; Sestini et al., 2013), computational and geometric methods, with particular emphasis on the flexibility and algebraic computability of PH curves (Habib and Sakai, 2007; Kong et al., 2008, 2012). (See Kim et al., 2007.)
We show how to find four generic interpolants to a $C^{1}$ Hermite data-set in the complex representation, using Pythagorean-hodograph curves generated as cuts of degree $(1, 3)$ of Laurent series. The developed numerical experiments have shown that two of these interpolants are simple curves and that these (at least) have stable shape, in the sense that their topologies persist when the direction of the velocity at each end-point changes. Our curves are fair, but have different shapes to those of other interpolants. Unlike existing methods, our technique allows regular PH interpolants to be found for special collinear $C^{1}$ Hermite data-sets.
A fully data-dependent criterion for free angles selection in spatial PH cubic biarc Hermite interpolation
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Citation Excerpt :
In the space the problem can be treated with PH quintics as well and in this case the selection of good interpolants becomes more challenging because there exists a two-parameter family of solutions, as first shown in Farouki et al. (2002). Among the other criteria for the selection of the free parameters, we mention the so called CC (Cubic Cubic) criterion firstly introduced in Farouki et al. (2008) and later better analyzed in Sestini et al. (2013) and in Farouki et al. (in press). Some numerical experiments that demonstrate the important features of our scheme are presented and the comparison with the interpolants obtained by using the angle selection proposed in Bastl et al. (2014) is given.
Cubic biarcs are the natural counterpart, in the context of the PH spatial Hermite interpolation, of piecewise quadratics for Hermite interpolation of standard parametric curves. Recently, it has been shown that, spatial Hermite interpolation can be always treated with PH cubic biarcs, providing an interesting alternative to the use of higher degrees, see Bastl et al. (2014). Beside the parameter value at the joint of the two arcs, which is typically free in any scheme of this kind, two free angular parameters remain. They should be automatically fixed by some suitable criterion in order to ensure the generation of interpolants with a reasonable shape. In this paper, an alternative to the constant choice of free parameters presented in Bastl et al. (2014) is proposed. It consists of an extension of the so called CC selection strategy introduced in Farouki et al. (2008) to the biarc setting. Such strategy is fully data-dependent, does not require any special configuration of the coordinate system and it guarantees the PH cubic reproduction when such kind of the interpolant exists. Moreover, the obtained scheme possesses other two important features, i.e. it is third order accurate and gives the possibility to control the torsion sign of the produced interpolant, in case with the introduction of an additional tension parameter relaxing the smoothness from $C^{1}$ to $G^{1}$ . The numerical results, based also on the straightforward spline extension of the scheme, confirm the developed theoretical analysis.
C2 Hermite interpolation by Pythagorean-hodograph quintic triarcs
2014, Computer Aided Geometric Design
In this paper, the problem of $C^{2}$ Hermite interpolation by triarcs composed of Pythagorean-hodograph (PH) quintics is considered. The main idea is to join three arcs of PH quintics at two unknown points – the first curve interpolates given $C^{2}$ Hermite data at one side, the third one interpolates the same type of given data at the other side and the middle arc is joined together with $C^{2}$ continuity to the first and the third arc. For any set of $C^{2}$ planar boundary data (two points with associated first and second derivatives) we construct four possible interpolants. The best possible approximation order is 4. Analogously, for a set of $C^{2}$ spatial boundary data we find a six-dimensional family of interpolating quintic PH triarcs. The results are confirmed by several examples.

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Abstract

Highlights

References (10)

Structural invariance of spatial Pythagorean hodographs

Comput. Aided Geom. Design

Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures

Comput. Aided Geom. Design

Clifford algebra, spin representation, and rational parameterization of curves and surfaces

Adv. Comp. Math.

Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable

Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves

Adv. Comp. Math.

Cited by (14)

Spatial quintic Pythagorean-hodograph interpolants to first-order Hermite data and Frenet frames

G<sup>1</sup> motion interpolation using cubic PH biarcs with prescribed length

Curvature continuous path planning and path finding based on PH splines with tension

Planar C<sup>1</sup> Hermite interpolation with PH cuts of degree (1, 3) of Laurent series

A fully data-dependent criterion for free angles selection in spatial PH cubic biarc Hermite interpolation

C<sup>2</sup> Hermite interpolation by Pythagorean-hodograph quintic triarcs