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Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation

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Abstract

In this article we define a new class of Pythagorean-Hodograph curves built-upon a six-dimensional mixed algebraic-trigonometric space, we show their fundamental properties and compare them with their well-known quintic polynomial counterpart. A complex representation for these curves is introduced and constructive approaches are provided to solve different application problems, such as interpolating C 1 Hermite data and constructing spirals as G 2 transition elements between a line segment and a circle, as well as between a pair of external circles.

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References

  1. Carnicer, J.M., Pena, J.M.: Totally positive bases for shape preserving curve design and optimality of B-splines. Comput. Aided Geom. Des. 11, 633–654 (1994)

    Article  MathSciNet  Google Scholar 

  2. Carnicer, J.M., Mainar, E., Pena, J.M.: Critical length for design purposes and extended Chebyshev spaces. Constr. Approx. 20, 55–71 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Farouki, R.T., Sakkalis, T.: Pythagorean hodographs. IBM J. Res. Dev. 34, 736–752 (1990)

    Article  MathSciNet  Google Scholar 

  4. Farouki, R.T.: The conformal map z → z 2 of the hodograph plane. Comput. Aided Geom. Des. 11, 363–390 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Farouki, R.T., Neff, C.A.: Hermite interpolation by pythagorean hodograph quintics. Math. Comput. 64(212), 1589–1609 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Farouki, R.T.: Pythagorean-hodograph quintic transition curves of monotone curvature. Comput. Aided Des. 29(9), 601–606 (1997)

    Article  Google Scholar 

  7. Farouki, R.T., Moon, H.P., Ravani, B.: Minkowski geometric algebra of complex sets. Geom. Dedicata 85, 283–315 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Farouki, R.T., Moon, H.P., Ravani, B.: Algorithms for Minkowski products and implicitly-defined complex sets. Adv. Comput. Math. 13, 199–229 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Farouki, R.T.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008)

    Book  Google Scholar 

  10. Guggenheimer, H.: Differential Geometry. McGrawHill, New York (1963)

    MATH  Google Scholar 

  11. Habib, Z., Sakai, M.: Transition between concentric or tangent circles with a single segment of G 2 PH quintic curve. Comput. Aided Geom. Des. 25, 247–257 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Moon, H.P., Farouki, R.T., Choi, H.I.: Construction and shape analysis of PH quintic Hermite interpolants. Comput. Aided Des. 18, 93–115 (2001)

    MATH  MathSciNet  Google Scholar 

  13. Mainar, E., Peña, J.M.: Corner cutting algorithms associated with optimal shape preserving representations. Comput. Aided Geom. Des. 16, 883–906 (1999)

    Article  MATH  Google Scholar 

  14. Mainar, E., Peña, J.M., Sánchez-Reyes, J.: Shape preserving alternatives to the rational Bézier model. Comput. Aided Geom. Des. 18, 37–60 (2001)

    Article  MATH  Google Scholar 

  15. Mainar, E., Peña, J.M.: A general class of Bernstein-like bases. Comput. Math. Appl. 53, 1686–1703 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mainar, E., Peña, J.M.: Optimal bases for a class of mixed spaces and their associated spline spaces. Comput. Math. Appl. 59, 1509–1523 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sánchez-Reyes, J.: Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials. Comput. Aided Geom. Des. 15, 909–923 (1998)

    Article  MATH  Google Scholar 

  18. Sasaki, T., Suzuki, M.: Three new algorithms for multivariate polynomial GCD. J. Symb. Comput. 13, 395–411 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dip. di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi 53, 20125, Milan, Italy

    Lucia Romani

  2. UNIV Lille Nord de France, UVHC, LAMAV-CGAO, FR no.2956, 59313, Valenciennes, France

    Laura Saini & Gudrun Albrecht

Authors
  1. Lucia Romani
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  2. Laura Saini
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  3. Gudrun Albrecht
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Correspondence to Lucia Romani.

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Communicated by: Rida T. Farouki

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Romani, L., Saini, L. & Albrecht, G. Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation. Adv Comput Math 40, 977–1010 (2014). https://doi.org/10.1007/s10444-013-9338-8

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  • DOI: https://doi.org/10.1007/s10444-013-9338-8

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