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Generating planar spiral by geometry driven subdivision scheme

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Abstract

Spirals are curves with one-signed, monotone increasing or decreasing curvature. They are commonly useful in a variety of applications, either for aesthetic or for engineering requirements. In this paper we propose a new iterative subdivision scheme for generating planar spiral segments from two points and their tangent vectors. The subdivision process consists of two main steps, computing new points and adjusting tangent vectors adaptively for each iteration. We categorize this iterative scheme as geometry driven because we utilize the old points and their tangent vectors whereas most other iterative schemes rely only on the old points. Some numerical examples are presented to show the advantageous properties of the new scheme.

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References

  1. Farin G. Curves and Surfaces for Computer Aided Geometric Design: A practical Guide. 4 ed. San Diego: Academic Press, 1997

    MATH  Google Scholar 

  2. Gibreel G M, Easa S M, Hassan Y, et al. State of the art of highway geometric design consistency. J Transp Eng, 1999, 125(4): 305–313

    Article  Google Scholar 

  3. Burchard H G, Ayers J A, Frey W H, et al. Approximation with aesthetic constraints. In: Designing Fair Curves and Surfaces, SIAM, Philadelphia, 1994

    Google Scholar 

  4. Mineur Y, Lichah T, Castelain J M, et al. A shape controlled fitting method for Bézier curves. Comput Aided Geom D, 1998, 15(9): 879–891

    Article  MATH  MathSciNet  Google Scholar 

  5. Frey W H, Field D A. Designing Bézier conics segments with monotone curvature. Comput Aided Geom D, 2000, 17(6): 457–483

    Article  MATH  MathSciNet  Google Scholar 

  6. Dietz D A, Piper B. Interpolation with cubic spirals. Comput Aided Geom D, 2004, 21(2): 165–180

    Article  MATH  MathSciNet  Google Scholar 

  7. Dietz D A, Piper B, Sebe E. Rational cubic spirals. Comput Aided Design, 2008, 40(1): 3–12

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Walton D J, Meek D S. A Pythagorean hodograph quintic spiral. Comput Aided Design, 1996, 28(12): 943–950

    Article  Google Scholar 

  10. Walton D J, Meek D S. A generalization of the Pythagorean hodograph quintic spiral. J Comput Appl Math, 2004, 172(2): 271–287

    Article  MATH  MathSciNet  Google Scholar 

  11. Habib Z, Sakai M. G 2 PH quintic spiral transition curves and their applications. Sci Math Japon, 2005, 61(2): 207–217

    MATH  MathSciNet  Google Scholar 

  12. Habib Z, Sakai M. On PH quintic spirals joining two circles with one circle inside the other. Comput Aided Design, 2007, 39(2): 125–132

    Article  Google Scholar 

  13. Habib Z, Sakai M. G 2 Pythagorean hodograph quintic transition between two circles with shape control. Comput Aided Geom D, 2007, 24(5): 252–266

    Article  MathSciNet  Google Scholar 

  14. Walton D J, Meek D S. Planar G 2 transition with a fair Pythagorean Hodograph quintic curve. J Comput Appl Math, 2002, 138(1): 109–126

    Article  MATH  MathSciNet  Google Scholar 

  15. Walton D J, Meek D S. G 2 curve design with a pair of Pythagorean Hodograph quintic spiral segments. Comput Aided Geom D, 2007, 24(5): 267–285

    Article  MathSciNet  Google Scholar 

  16. Yang X. Normal based subdivision scheme for curve design. Comput Aided Geom D, 2006, 23(3): 243–260

    Article  MATH  Google Scholar 

  17. Dyn N, Levin D, Gregory J A. A 4-point interpolatory subdivision scheme for curve design. Comput Aided Geom D, 1987, 4(4): 257–268

    Article  MATH  MathSciNet  Google Scholar 

  18. Dyn N, Floater M S, Hormann K. A C 2 four-point subdivision scheme with fourth order accuracy and its extensions. In: Daehlen M, Mørken K, Schumaker L L, eds. Mathematical Methods for Curves and Surfaces, Tromso 2004, 2005. 145–156

  19. Hassan M F, Ivrissimitzis I P, Dodgson N A, et al. An interpolating 4-point C 2 ternary stationary subdivision scheme. Comput Aided Geom D, 2002, 19(1): 1–18

    Article  MATH  MathSciNet  Google Scholar 

  20. Marinov M, Dyn N, Levin D. Geometrically controlled 4-point interpolatory schemes. In: Dodgson N A, Floater M S, Sabin M A, eds. Advances in Multiresolution for Geometric Modeling. Berlin: Springer-Verlag, 2005

    Google Scholar 

  21. Aspert N, Ebrahimi T, Vandergheynst P. Non-linear subdivision using local spherical coordinates. Comput Aided Geom D, 2003, 20(3): 165–187

    Article  MATH  MathSciNet  Google Scholar 

  22. Guggenheimer H. Differential Geometry. New York: McGraw-Hill, 1963

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    ChongYang Deng & GuoZhao Wang

  2. Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Hangzhou, 310018, China

    ChongYang Deng

Authors
  1. ChongYang Deng
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  2. GuoZhao Wang
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Corresponding author

Correspondence to ChongYang Deng.

Additional information

Supported partially by the National Natural Science Foundation of China (Grant Nos. 60673032, 60773179), and the National Basic Research Program of China (Grant No. 2004CB318000), and the Scientific Starting Foundation of Hangzhou Dianzi University

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Deng, C., Wang, G. Generating planar spiral by geometry driven subdivision scheme. Sci. China Ser. F-Inf. Sci. 52, 1821–1829 (2009). https://doi.org/10.1007/s11432-009-0160-3

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  • DOI: https://doi.org/10.1007/s11432-009-0160-3

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