On the approximation order of a space data-dependent PH quintic Hermite interpolation scheme
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 spline extension of the scheme is considered for the experiments.
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Cited by (14)
Spatial quintic Pythagorean-hodograph interpolants to first-order Hermite data and Frenet frames
2021, Computer Aided Geometric DesignCitation Excerpt :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.
G<sup>1</sup> motion interpolation using cubic PH biarcs with prescribed length
2018, Computer Aided Geometric DesignCurvature continuous path planning and path finding based on PH splines with tension
2017, CAD Computer Aided DesignPlanar C<sup>1</sup> Hermite interpolation with PH cuts of degree (1, 3) of Laurent series
2014, Computer Aided Geometric DesignCitation 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.)
A fully data-dependent criterion for free angles selection in spatial PH cubic biarc Hermite interpolation
2014, Computer Aided Geometric DesignCitation 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.
C<sup>2</sup> Hermite interpolation by Pythagorean-hodograph quintic triarcs
2014, Computer Aided Geometric Design