Spiral transitions

  • Akin Levent
  • Bayram Sahin
  • Zulfiqar HabibEmail author


Spiral curves are free from singularities and curvature extrema. These are used in path smoothing applications to overcome the abrupt change in curvature and super-elevation of moving object that occurs between tangent and circular curve. Line to circle spiral transition is made of straight line segment and curvature continuous spiral curve. It is extendible to other important types of transitions like line to line and circle to circle. Although the problem of line to circle transition has been addressed by many researchers, there is no comprehensive literature review available. This paper presents state-of-the-art of line to circle spiral transition, applications in different fields, limitations of existing approaches, and recommendations to meet the challenges of compatibility with today’s CAD/CAM soft wares, satisfaction of Hermite end conditions, approximation of discrete data for image processing, 3D path smoothness for an unmanned aerial vehicle (UAV), and arc-length parametrization. Whole discussion is concluded with future research directions in various areas of applications.


path planning spiral continuity curvature extrema line to circle transition 

MR Subject Classification



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This research is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under the Visiting Scientist Programme; and PDE-GIR project which has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie grant agreement No 778035


  1. [1]
    Benjamin Adler, Junhao Xiao, Jianwei Zhang. Autonomous exploration of urban environ-ments using unmanned aerial vehicles, Journal of Field Robotics, 31(6):912–939, 2014.Google Scholar
  2. [2]
    Mohammad Al Khawaldah, Andreas Nüchter. Enhanced frontier-based exploration for in-door environment with multiple robots, Advanced Robotics, 29(10):657–669, 2015.Google Scholar
  3. [3]
    Gianluca Antonelli, G Antonelli. Underwater robots, Springer, 2014.zbMATHGoogle Scholar
  4. [4]
    Khurshid Asghar, Zulfiqar Habib, Muhammad Hussain. Copy-move and splicing image forgery detection and localization techniques: a review, Australian Journal of Forensic Sciences, pages 1–27, 2016.Google Scholar
  5. [5]
    K G Baass. The use of clothoid templates in highway design, Transportation Forum, 1:47–52, 1984.Google Scholar
  6. [6]
    Jose Luis Blanco, Mauro Bellone, Antonio Gimenez-Fernandez. Tp-space rrt-kinematic path planning of non-holonomic any-shape vehicles, International Journal of Advanced Robotic Systems, 12, 2015.Google Scholar
  7. [7]
    David BL Bong, Koon Chun Lai, Annie Joseph. Automatic road network recognition and extraction for urban planning, International Journal of Applied Science, Engineering and Technology, 5(1):209–215, 2009.Google Scholar
  8. [8]
    JL Campbell, CM Richard, J Graham. Human factors guidelines for roadway systems, washington, dc. Technical report, NCHRP Report, 2008.Google Scholar
  9. [9]
    Anjan Chakrabarty, Jack W Langelaan. Energy-based long-range path planning for soaring-capable unmanned aerial vehicles, Journal of Guidance, Control, and Dynamics, 34(4):1002–1015, 2011.Google Scholar
  10. [10]
    Jie Chen, Guo-Jin Wang. A new type of the generalized bézier curves, Applied Mathematics-A Journal of Chinese Universities, 26(1):47–56, 2011.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Yonghua Chen, Fenghua Dong. Robot machining: recent development and future research issues, The International Journal of Advanced Manufacturing Technology, 66(9-12):1489–1497, 2013.Google Scholar
  12. [12]
    Howie M Choset. Principles of robot motion: theory, algorithms, and implementation, MIT press, 2005.zbMATHGoogle Scholar
  13. [13]
    Licai Chu, Xiao-Ming Zeng. Constructing curves and triangular patches by beta functions, Journal of Computational and Applied Mathematics, 260:191–200, 2014.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Dominique David, Nathalie Sprynski. Method and device for acquisition of a geometric shape, November 17 2015. US Patent 9,188,422.Google Scholar
  15. [15]
    D A Dietz, B Piper, E Sebe. Rational cubic spirals, Computer-Aided Design, 40:3–12, 2008.Google Scholar
  16. [16]
    D W Dudley. Handbook of practical gear design, CRC Press, 1994.Google Scholar
  17. [17]
    Mohamed Elbanhawi, Milan Simic, Reza N Jazar. Continuous path smoothing for car-like robots using b-spline curves, Journal of Intelligent & Robotic Systems, 80(1):23–56, 2015.Google Scholar
  18. [18]
    Gerald Farin. Curves, Surfaces for CAGD: A Practical Guide. Morgan-Kaufmann, 5th edition, 2002.Google Scholar
  19. [19]
    Gerald Farin. Geometric Hermite interpolation with circular precision, Computer-Aided Design, 40(4):476–479, 2008.Google Scholar
  20. [20]
    Rida T Farouki. Construction of G1 planar hermite interpolants with prescribed arc lengths, Computer Aided Geometric Design, 2016.Google Scholar
  21. [21]
    R T Farouki, T Sakkalis. Pythagorean hodographs, IBM Journal of Research and Development, 34(5):736–752, 1990.MathSciNetGoogle Scholar
  22. [22]
    G M Gibreel, S M Easa, Y Hassan, I A El-Dimeery. State of the art of highway geometric design consistency, ASCE Journal of Transportation Engineering, 125(4):305–313, 1999.Google Scholar
  23. [23]
    T N T Goodman, D S Meek. Planar interpolation with a pair of rational spirals, Computational and Applied Mathematics, 201:112–127, 2007.MathSciNetzbMATHGoogle Scholar
  24. [24]
    T N T Goodman, B H Ong, K Unsworth. NURBS for Curve and Surface Design, chapter Constrained Interpolation Using Rational Cubic Splines, pages 59–74. SIAM, Philadelphia, 1991.Google Scholar
  25. [25]
    H Guggenheimer. Differential Geometry, McGraw-Hill, New York, 1963.zbMATHGoogle Scholar
  26. [26]
    D Guo, A Weeks, H Klee. Robust approach for suburban road segmentation in high-resolution aerial images, International Journal of Remote Sensing, 28(2):307–318, 2007.Google Scholar
  27. [27]
    Zulfiqar Habib. Spiral Function and Its Application in CAGD, VDM Verlag, Germany, 2010. ISBN: 978-3-639-24988-0.Google Scholar
  28. [28]
    Zulfiqar Habib, Manabu Sakai. G2 two-point Hermite rational cubic interpolation, International Journal of Computer Mathematics, 79(11):1225–1231, 2002. Scholar
  29. [29]
    Zulfiqar Habib, Manabu Sakai. G2 planar cubic transition between two circles, International Journal of Computer Mathematics, 80(8):959–967, 2003. Scholar
  30. [30]
    Zulfiqar Habib, Manabu Sakai. Advances in Geometric Modeling, chapter Family of G2 Spiral Transition Between Two Circles, pages 133–150, John Wiley, January 2004, ISBN: 0-470-85937-7,,descCd-tableOfContents.html.Google Scholar
  31. [31]
    Zulfiqar Habib, Manabu Sakai. Shapes of planar cubic curves, Scientiae Mathematicae Japonicae, 59(1):133–138, 2004.:e9, 253–258, Scholar
  32. [32]
    Zulfiqar Habib, Manabu Sakai. Spiral transition curves and their applications, Scientiae Mathematicae Japonicae, 61(2):195–206, 2005. e2004, 251–262, Scholar
  33. [33]
    Zulfiqar Habib, Manabu Sakai. G2 Pythagorean hodograph quintic transition between t-wo circles with shape control, Computer Aided Geometric Design, 24(5):252–266, 2007. Scholar
  34. [34]
    Zulfiqar Habib, Manabu Sakai. On PH quintic spirals joining two circles with one circle inside the other, Computer Aided Design, 39(2):125–132, 2007. Scholar
  35. [35]
    Zulfiqar Habib, Manabu Sakai. Transition between concentric or tangent circles with a single segment of G2 PH quintic curve, Computer Aided Geometric Design, 25(4-5):247–257, 2008. Scholar
  36. [36]
    Zulfiqar Habib, Manabu Sakai. G2 cubic transition between two circles with shape control, Computational and Applied Mathematics, 223:133–144, 2009, Scholar
  37. [37]
    Zulfiqar Habib, Manabu Sakai. Admissible regions for rational cubic spirals matching G2 Hermite data, Computer-Aided Design, 42(12):1117–1124, 2010, Scholar
  38. [38]
    Zulfiqar Habib, Manabu Sakai. Cubic spiral transition matching G2 Hermite end condi-tions, Numerical Mathematics, Theory, Methods, Applications, 4(4):525–536, 2011.MathSciNetGoogle Scholar
  39. [39]
    Zulfiqar Habib, Manabu Sakai. Fairing arc spline and designing by using cubic Bézier spiral segments, Mathematical Modelling and Analysis, 17(2):141–160, 2012.MathSciNetzbMATHGoogle Scholar
  40. [40]
    Zulfiqar Habib, Manabu Sakai. Fairing an arc spline and designing with G2 PH quintic spiral transitions, International Journal of Computer Mathematics, 90(5):1023–1039, 2013.MathSciNetzbMATHGoogle Scholar
  41. [41]
    Zulfiqar Habib, Muhammad Sarfraz, Manabu Sakai. Rational cubic spline interpolation with shape control, Computers & Graphics, 29(4):594–605, 2005, Scholar
  42. [42]
    P Hartman. The highway spiral for combining curves of different radii, Transactions of the American Society of Civil Engineers, 122:389–409, 1957.Google Scholar
  43. [43]
    P Henrici. Applied and Computational Complex Analysis, volume 1. Wiley, New York, 1988.Google Scholar
  44. [44]
    Josef Hoschek, Dieter Lasser. Fundamentals of Computer Aided Geometric Design (Trans-lation by L.L. Schumaker). A, K. Peters, Wellesley, MA, 1993.zbMATHGoogle Scholar
  45. [45]
    Md Arafat Hossain, Israt Ferdous. Autonomous robot path planning in dynamic environ-ment using a new optimization technique inspired by bacterial foraging technique, Robotics and Autonomous Systems, 64:137–141, 2015.Google Scholar
  46. [46]
    Mathieu Huard, Rida T Farouki, Nathalie Sprynski, Luc Biard. C2 interpolation of s-patial data subject to arc-length constraints using pythagorean hodograph quintic splines, Graphical models, 76(1):30–42, 2014.Google Scholar
  47. [47]
    Uk-Youl Huh, Seong-Ryong Chang. A G2 continuous path-smoothing algorithm using mod-ified quadratic polynomial interpolation, International Journal of Advanced Robotic Systems, 11(25), 2014.Google Scholar
  48. [48]
    KG Jolly, R Sreerama Kumar, R Vijayakumar. A bezier curve based path planning in a multi-agent robot soccer system without violating the acceleration limits, Robotics and Autonomous Systems, 57(1):23–33, 2009.Google Scholar
  49. [49]
    Amna Khan, IRAM NOREEN, Zulfiqar Habib. On complete coverage path planning al-gorithms for non-holonomic mobile robots: Survey and challenges. Journal of Information Science & Engineering, 33(1), 2017.Google Scholar
  50. [50]
    Amna Khan, Iram Noreen, Hyejeong Ryu, Nakju Lett Doh, Zulfiqar Habib. Online com-plete coverage path planning using two-way proximity search, Intelligent Service Robotics, pages 1–12, 2017.Google Scholar
  51. [51]
    Steven M LaValle. Planning algorithms, Cambridge university press, 2006.zbMATHGoogle Scholar
  52. [52]
    Raphael Linus Levien. From spiral to spline: Optimal techniques in interactive curve de-sign, University of California, Berkeley, 2009.Google Scholar
  53. [53]
    Z Li, D S Meek. Smoothing an arc spline, Computers & Graphics, 29:576–587, 2005.Google Scholar
  54. [54]
    Zhong Li, Lizhuang Ma, Mingxi Zhao, Zhihong Mao. Improvement construction for planar G2 transition curve between two separated circles, In International Conference on Computational Science, pages 358–361. Springer, 2006.Google Scholar
  55. [55]
    T C Liang, J S Liu, G T Hung, Y Z Chang. Practical and flexible path planning for car-like mobile robot using maximal-curvature cubic spiral, Robotics and Autonomous Systems, 52(4):312–335, 2005.Google Scholar
  56. [56]
    Chengkai Lu, Lizheng Jiang. An iterative algorithm for G2 multiwise merging of Bézier curves, Journal of Computational and Applied Mathematics, pages 352–361, 2016.Google Scholar
  57. [57]
    Lizheng Lu. Planar quintic G2 hermite interpolation with minimum strain energy, Journal of Computational and Applied Mathematics, 274:109–117, 2015.MathSciNetzbMATHGoogle Scholar
  58. [58]
    Ellips Masehian, Davoud Sedighizadeh. Multi-objective robot motion planning using a parti-cle swarm optimization model, Journal of Zhejiang University SCIENCE C, 11(8):607–619, 2010.Google Scholar
  59. [59]
    Asif Masood, Muhammad Sarfraz. An efficient technique for capturing 2d objects, Computers & Graphics, 32(1):93–104, 2008.Google Scholar
  60. [60]
    D S Meek, D J Walton. The use of cornu spirals in drawing planar curves of controlled curvature, Computational and Applied Mathematics, 25:69–78, 1989.MathSciNetzbMATHGoogle Scholar
  61. [61]
    D S Meek, D J Walton. The use of Cornu spirals in drawing planar curves of controlled curvature, Computational and Applied Mathematics, 25:69–78, 1989.MathSciNetzbMATHGoogle Scholar
  62. [62]
    Dereck S Meek, Desmond J Walton. Approximation of discrete data by G1 arc splines, Computer-Aided Design, 24(6):301–306, 1992.zbMATHGoogle Scholar
  63. [63]
    Ana Margarida Monteiro, Reha H Tütüncü, Luís N Vicente. Recovering risk-neutral proba-bility density functions from options prices using cubic splines and ensuring nonnegativity, European Journal of Operational Research, 187(2):525–542, 2008.MathSciNetzbMATHGoogle Scholar
  64. [64]
    Guochen Niu, Lili Liu, Hao Chen, Qingji Gao. Set-point stabilization of nonholonomic mobile robot based on optimizing bezier curve, In Intelligent Control and Automation, 2008. WCICA 2008. 7th World Congress on, pages 1609–1612. IEEE, 2008.Google Scholar
  65. [65]
    Iram Noreen, Amna Khan, Zulfiqar Habib. A comparison of rrt, rrt* and rrt*-smart path planning algorithms, International Journal of Computer Science and Network Security (IJCSNS), 16(10):20, 2016.Google Scholar
  66. [66]
    Iram Noreen, Amna Khan, Zulfiqar Habib. Optimal path planning using rrt* based ap-proaches: A survey and future directions, International Journal of Advanced Computer Science & Applications, 1(7):97–107, 2016.Google Scholar
  67. [67]
    M Pan, X Yang, J Tang. Research on interpolation methods in medical image processing, Journal of Medical Systems, 36(2):777–807, July 2012.Google Scholar
  68. [68]
    L Piegl, W Tiller. The NURBS book, Springer, 1995.zbMATHGoogle Scholar
  69. [69]
    M Rajeswari, KS Gurumurthy, L Pratap Reddy, SN Omkar, J Senthilnath. Automatic road extraction based on normalized cuts and level set methods, Int J Comput App, 18:10–16, 2011.Google Scholar
  70. [70]
    Dimulyo Sarpono, Zulfiqar Habib, Manabu Sakai. Fair cubic transition between two circles with one circle inside or tangent to the other, Numerical Algorithms, 51(4):461–476, July 2009. Scholar
  71. [71]
    Dimulyo Sarpono, Zulfiqar Habib, Manabu Sakai. Fair cubic transition between two circles with one circle inside or tangent to the other, Numerical Algorithms, 51(4):461–476, July 2009. Scholar
  72. [72]
    D J Walton, D S Meek. Computer-aided design for horizontal alignment, ASCE Journal of Transporation Engineering, 115:411–424, 1989.Google Scholar
  73. [73]
    D J Walton, D S Meek. Clothoid splines, Computers & Graphics, 14:95–100, 1990.Google Scholar
  74. [74]
    D J Walton, D S Meek. A planar cubic Bézier spiral, Computational and Applied Mathematics, 72(1):85–100, 1996.MathSciNetzbMATHGoogle Scholar
  75. [75]
    D J Walton, D S Meek. A Pythagorean hodograph quintic spiral, Computer-Aided Design, 28(12):943–950, 1996.Google Scholar
  76. [76]
    D J Walton, D S Meek. G2 curves composed of planar cubic and Pythagorean hodograph quintic spirals, Computer Aided Geometric Design, 15(6):547–566, 1998.MathSciNetzbMATHGoogle Scholar
  77. [77]
    D J Walton, D S Meek. Planar G2 transition between two circles with a fair cubic Bézier curve, Computer-Aided Design, 31(14):857–866, 1999.zbMATHGoogle Scholar
  78. [78]
    D J Walton, D S Meek. Planar G2 transition with a fair Pythagorean hodograph quintic curve, Computational and Applied Mathematics, 138:109–126, 2002.MathSciNetzbMATHGoogle Scholar
  79. [79]
    D J Walton, D S Meek. A controlled clothoid spline, Computers & Graphics, 29:353–363, 2005.Google Scholar
  80. [80]
    D J Walton, D S Meek. G2 curve design with a pair of Pythagorean hodograph quintic spiral segments, Computer Aided Geometric Design, 24(5):267–285, 2007.MathSciNetzbMATHGoogle Scholar
  81. [81]
    D J Walton, D S Meek. A further generalisation of the planar cubic Bézier spiral, Computational and Applied Mathematics, 236(11):2869–2882, 2012.MathSciNetzbMATHGoogle Scholar
  82. [82]
    D J Walton, D S Meek, J M Ali. Planar G2 transition curves composed of cubic Bézier spiral segments, Computational and Applied Mathematics, 157(2):453–476, 2003.MathSciNetzbMATHGoogle Scholar
  83. [83]
    H Wang, J Kearney, K Atkinson. Arc-length parameterized spline curves for real-time simulation, In Curve and Surface Design, pages 387–396, San Malo, France, June 2002. Proceedings of the 5th International Conference on Curves and Surfaces, Nashboro Press.
  84. [84]
    Joab R Winkler. A unified approach to resultant matrices for bernstein basis polynomials, Computer Aided Geometric Design, 25(7):529–541, 2008.MathSciNetzbMATHGoogle Scholar
  85. [85]
    Weidong Wu, Xunnian Yang. Geometric hermite interpolation by a family of intrinsically defined planar curves, Computer-Aided Design, 2016.Google Scholar
  86. [86]
    Lianghong Xu, Jianhong Shi. Geometric hermite interpolation for space curves, Computer aided geometric design, 18(9):817–829, 2001.MathSciNetzbMATHGoogle Scholar
  87. [87]
    J Yang, RS Wang. Classified road detection from satellite images based on perceptual or-ganization, International Journal of Remote Sensing, 28(20):4653–4669, 2007.Google Scholar
  88. [88]
    K Yang, S K Gan, S Sukkarieh. An efficient path planning and control algorithm for RUAV’s in unknown and cluttered environmnets, Journal of Intelligent Robot System, 57:101–122, 2010.zbMATHGoogle Scholar
  89. [89]
    Kwangjin Yang, Sangwoo Moon, Seunghoon Yoo, Jaehyeon Kang, Nakju Lett Doh, Hong Bong Kim, Sanghyun Joo. Spline-based rrt path planner for non-holonomic robots, Journal of Intelligent & Robotic Systems, 73(1-4):763–782, 2014.Google Scholar
  90. [90]
    Kwangjin Yang, Salah Sukkarieh. An analytical continuous-curvature path-smoothing al-gorithm, IEEE Transactions on Robotics, 26(3):561–568, 2010.Google Scholar
  91. [91]
    Jun-Hai Yong, Shi-Min Hu, Jia-Guang Sun. A note on approximation of discrete data by G1 arc splines, Computer-Aided Design, 31(14):911–915, 1999.zbMATHGoogle Scholar
  92. [92]
    Junku Yuh. Design and control of autonomous underwater robots: A survey, Autonomous Robots, 8(1):7–24, 2000.Google Scholar
  93. [93]
    Beong In Yun. A cumulative averaging method for piecewise polynomial approximation to discrete data, Applied Mathematical Sciences, 10(7):331–343, 2016.Google Scholar
  94. [94]
    Zhi-hao Zheng, Guo-zhao Wang. A note on pythagorean hodograph quartic spiral, Applied Mathematics-A Journal of Chinese Universities, 33(2):234–252, 2018.MathSciNetzbMATHGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of ScienceInonu UniversityMalatyaTurkey
  2. 2.Faculty of ScienceEge UniversityIzmirTurkey
  3. 3.Department of Computer ScienceCOMSATS University IslamabadLahore CampusPakistan

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