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Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

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Abstract

It is shown that, depending upon the orientation of the end tangents t0,t1 relative to the end point displacement vector Δp=p1−p0, the problem of G1 Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points p0,. . .,pn in R3, compatible with a G1 piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples.

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Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Belzoni 7, 35131, Padova, Italy

    Francesca Pelosi

  2. Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA, 95616, USA

    Rida T. Farouki

  3. Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133, Roma, Italy

    Carla Manni

  4. Dipartimento di Energetica, Università di Firenze, via Lombroso 6/17, 50134, Firenze, Italy

    Alessandra Sestini

Authors
  1. Francesca Pelosi
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  2. Rida T. Farouki
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  3. Carla Manni
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  4. Alessandra Sestini
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Corresponding author

Correspondence to Francesca Pelosi.

Additional information

Communicated by C.A. Micchelli

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Pelosi, F., Farouki, R.T., Manni, C. et al. Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics. Adv Comput Math 22, 325–352 (2005). https://doi.org/10.1007/s10444-003-2599-x

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  • DOI: https://doi.org/10.1007/s10444-003-2599-x

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