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A New Method to Design Cubic Pythagorean-Hodograph Spline Curves with Control Polygon

  • Hongmei KangEmail author
  • Xin Li
Article
  • 27 Downloads

Abstract

A new method to design a cubic Pythagorean-hodograph (PH) spline curve from any given control polygon is proposed. The key idea is to suitably choose a set of auxiliary points associated with the edges of the given control polygon to guarantee the constructed PH spline has \(G^1\) continuity or curvature continuity. The method facilitates intuitive and efficient construction of open and closed cubic PH spline curves that typically agrees closely with the same friendly interface and properties as B-splines, for example, the convex hull and variation-diminishing properties.

Keywords

Cubic pythagorean-hodograph (PH) curve Control polygon Interactive design \(G^1\) continuity Curvature continuity 

Mathematics Subject Classification

65D07 65D17 

References

  1. 1.
    Albrecht, G., Beccari, C.V., Canonne, J.-C., Romani, L.: Planar Pythagorean-hodograph B-spline curves. Comput. Aided Geom. Des. 57, 57–77 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Albrecht, G., Farouki, R.T.: Construction of \(C^2\) Pythagorean hodograph interpolating splines by the homotopy method. Adv. Comp. Math. 5, 417–442 (1996)CrossRefGoogle Scholar
  3. 3.
    Choi, H.I., Farouki, R.T., Kwon, S.-H., Moon, H.P.: Topological criterion for selection of quintic Pythagorean-hodograph hermite interpolants. Comput. Aided Geom. Des. 25, 411–433 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Choi, H.I., Han, C.Y., Moon, H.P., Roh, K.H., Wee, N.-S.: Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves. Comput. Aided Des. 31, 59–72 (1999)CrossRefGoogle Scholar
  5. 5.
    Dong, B., Farouki, R.T.: PHquintic: a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph curves. ACM Trans. Math. Softw. (TOMS) (2015).  https://doi.org/10.1145/2699467
  6. 6.
    Farouki, R.: Pythagorean hodograph curves in practical use. In: Barnhill, R.E. (ed.) Geometry Processing for Design and Manufacturing, pp. 3–33. SIAM, Philadelphia (1992)CrossRefGoogle Scholar
  7. 7.
    Farouki, R.: Pythagorean-hodograph curves. In: Farin, G., Hoschek, J., Kim, M.-S. (eds.) Handbook of Computer Aided Geometric Design, pp. 405–427. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar
  8. 8.
    Farouki, R.T.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008)Google Scholar
  9. 9.
    Farouki, R., Manjunathaiah, J., Nichlas, D., Yuan, G., Jee, S.: Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves. Comput. Aided Des. 30, 631–640 (1998)CrossRefGoogle Scholar
  10. 10.
    Farouki, R., Manni, C., Pelosi, F., Sampoli, M.: Design of \(C^2\) spatial Pythagorean-hodograph quintic spline curves by control polygons. In: Lecture Notes in Computer Science, vol. 6920, pp. 253–269 (2012)Google Scholar
  11. 11.
    Farouki, R., Neff, C.: Hermite interpolation by Pythagorean hodograph quintics. Math. Comput. 64, 1589–1609 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Farouki, R., Sakkalis, T.: Pythagorean hodographs. IBM J. Res. Dev. 34, 736–752 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Farouki, R., Sakkalis, T.: Pythagorean-hodograph space curves. Adv. Comput. Math. 2, 41–66 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Farouki, R., Shah, S.: Real-time CNC interpolators for Pythagorean-hodograph curves. Comput. Aided Geom. Des. 13, 583–600 (1996)CrossRefGoogle Scholar
  15. 15.
    Farouki, R.T.: Exact rotation-minimizing frames for spatial Pythagorean hodograph curves. Gr. Models 64, 382–395 (2002)CrossRefGoogle Scholar
  16. 16.
    Farouki, R.T., Giannelli, C., Sestini, A.: Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form. Adv. Comput. Math. 42, 199–225 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Farouki, R.T., Manni, C., Sestini, A.: Shape-preserving interpolation by \(G^1\) and \(G^2\) PH quintic splines. IMA J. Numer. Anal. 23, 175–195 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Farouki, R.T., Tsai, Y.-F., Yuan, G.-F.: Contour machining of free-form surfaces with real-time PH curve CNC interpolators. Comput. Aided Geom. Des. 16, 61–76 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jaklič, G., Kozak, J., Krajnc, M., Vitrih, V., Žagar, E.: On interpolation by planar cubic \(G^2\) Pythagorean-hodograph spline curves. Math. Comput. 79, 305–326 (2010)CrossRefGoogle Scholar
  20. 20.
    Jüttler, B.: Hermite interpolation by Pythagorean hodograph curves of degree seven. Math. Comput. 70, 1089–1111 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jüttler, B., Mäurer, C.: Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling. Comput. Aided Des. 31, 73–83 (1999)CrossRefGoogle Scholar
  22. 22.
    Mäurer, C., Jüttler, B.: Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics. J. Geom. Gr. 3, 141–159 (1999)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Meek, D., Walton, D.: Geometric Hermite interpolation with Tschirnhausen cubics. J. Comput. Appl. Math. 81, 299–309 (1997)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Moon, H., Farouki, R.T., Choi, H.: Construction and shape analysis of PH quintic hermite interpolants. Comput. Aided Geom. Des. 18, 93–115 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pelosi, F., Farouki, R.T., Manni, C., Sestini, A.: Geometric hermite interpolation by spatial Pythagorean hodograph cubics. Adv. Comput. Math. 22, 325–352 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pelosi, F., Sampoli, M., Farouki, R., Manni, C.: A control polygon scheme for design of planar \(C^2\) PH quintic spline curves. Comput. Aided Geom. Des. 24, 28–52 (2007)CrossRefGoogle Scholar
  27. 27.
    Tsai, Y.-F., Farouki, R.T., Feldman, B.: Performance analysis of CNC interpolators for time-dependent feedrates along PH curves. Comput. Aided Geom. Des. 18, 245–265 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, G., Fang, L.: On control polygons of quartic Pythagorean-hodograph curves. Comput. Aided Geom. Des. 26, 1006–1015 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Soochow UniversitySuzhouChina
  2. 2.University of Science and Technology of ChinaHefeiChina

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