Numerical Computation of Egg and Double-Egg Curves with Clothoids
Publication: Journal of Surveying Engineering
Volume 146, Issue 1
Abstract
The clothoid, also known as Cornu spiral or Euler spiral, is a curve widely used as a transition curve when designing the layout of railway tracks and roads because of a key feature: its curvature is proportional to its length. The classical method to compute a clothoid is based on the use of Taylor expansions of sine and cosine functions, usually starting with zero curvature at the initial point. In this paper the clothoid is presented as the only curve with a constant rate of change of curvature, which parametrization can be obtained by solving an initial value problem. In this initial value problem the curvature at the starting point can be chosen, being able to develop simple, efficient, and accurate algorithms to connect two oriented circumferences by means of clothoids. These algorithms are presented as a useful tool for designing egg and double-egg curves in highway connections and interchanges.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
M.E. Vázquez-Méndez thanks the funding support from project MTM2015-65570-P of Ministerio de Economía y Competitividad (Spain)/FEDER. J.B. Ferreiro thanks the financial support of Xunta de Galicia (Spain), project ED431C 2019/02.
References
Atkinson, K., W. Han, and D. Stewart. 2009. Numerical solution of ordinary differential equations. Hoboken, NJ: Wiley.
Baass, K. 1984. “Use of clothoid templates in highway design.” Transp. Forum 1 (3): 47–52.
Baykal, O., E. Tari, Z. Çoşkun, and M. Şahin. 1997. “New transition curve joining two straight lines.” J. Transp. Eng. 123 (5): 337–345. https://doi.org/10.1061/(ASCE)0733-947X(1997)123:5(337).
Bosurgi, G., and A. D’Andrea. 2012. “A polynomial parametric curve (PPC-CURVE) for the design of horizontal geometry of highways.” Comput.-Aided Civ. Infrastruct. Eng. 27 (4): 304–312. https://doi.org/10.1111/j.1467-8667.2011.00750.x.
Casal, G., D. Santamarina, and M. E. Vázquez-Méndez. 2017. “Optimization of horizontal alignment geometry in road design and reconstruction.” Transp. Res. Part C: Emerging Technol. 74 (Jan): 261–274. https://doi.org/10.1016/j.trc.2016.11.019.
Dong, H., S. M. Easa, and J. Li. 2007. “Approximate extraction of spiralled horizontal curves from satellite imagery.” J. Surv. Eng. 133 (1): 36–40. https://doi.org/10.1061/(ASCE)0733-9453(2007)133:1(36).
Easa, S. M., and Y. Hassan. 2000a. “Development of transitioned vertical curve. I: Properties.” Trans. Res. A 34 (6): 481–496.
Easa, S. M., and Y. Hassan. 2000b. “Development of transitioned vertical curve. II: Sight distance.” Trans. Res. A 34 (7): 565–584.
Heald, M. A. 1985. “Rational approximations for the fresnel integrals.” Math. Comput. 44 (170): 459–461. https://doi.org/10.1090/S0025-5718-1985-0777277-6.
Kobryń, A. 1993. “General mathematical transition curves for alignment between two rectilinear road sections.” Zeitschrift fur Vermessungswesen 5: 227–242.
Kobryń, A. 2011. “Polynomial solutions of transition curves.” J. Surv. Eng. 137 (3): 71–80. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000044.
Kobryń, A. 2014. “New solutions for general transition curves.” J. Surv. Eng. 140 (1): 12–21. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000113.
Kobryń, A. 2016a. “Universal solutions of transition curves.” J. Surv. Eng. 142 (4): 04016010. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000179.
Kobryń, A. 2016b. “Vertical arcs design using polynomial transition curves.” KSCE J. Civ. Eng. 20 (1): 376–384. https://doi.org/10.1007/s12205-015-0492-z.
Kobryń, A. 2017. Transition curves for highway geometric design. Vol. 14 of Springer tracts on transportation and traffic. Cham, Switzerland: Springer International Publishing.
Koç, I., K. Gümüş, and M. O. Selbesoğlu. 2015. “Design of double-egg curve in the link roads of transportation networks.” Tehnički vjesnik 22 (2): 495–501.
Lovell, D. J. 1999. “Automated calculation of sight distance from horizontal geometry.” J. Transp. Eng. 125 (4): 297–304. https://doi.org/10.1061/(ASCE)0733-947X(1999)125:4(297).
Lovell, D. J., J. Jong, and P. C. Chang. 2001. “Improvements to sight distance algorithm.” J. Transp. Eng. 127 (4): 283–288. https://doi.org/10.1061/(ASCE)0733-947X(2001)127:4(283).
Meek, D., and D. Walton. 1989. “The use of Cornu spirals in drawing planar curves of controlled curvature.” J. Comput. Appl. Math. 25 (1): 69–78. https://doi.org/10.1016/0377-0427(89)90076-9.
Meek, D., and D. Walton. 2004a. “A note on finding clothoids.” J. Comput. Appl. Math. 170 (2): 433–453. https://doi.org/10.1016/j.cam.2003.12.047.
Meek, D., and D. Walton. 2004b. “An arc spline approximation to a clothoid.” J. Comput. Appl. Math. 170 (1): 59–77. https://doi.org/10.1016/j.cam.2003.12.038.
Powell, M. J. D. 1970. “A Fortran subroutine for solving systems of nonlinear algebraic equations.” Chap. 7 in Numerical methods for nonlinear algebraic equations, edited by P. Rabinowitz. New York: Gordon and Breach.
Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. 2002. Numerical recipes in C: The art of scientific computing. 2nd ed. New York: Cambridge University Press.
Sánchez-Reyes, J., and J. Chacón. 2003. “Polynomial approximation to clothoids via s-power series.” Comput.-Aided Des. 35 (14): 1305–1313. https://doi.org/10.1016/S0010-4485(03)00045-9.
Stoer, J. 1982. “Curve fitting with clothoidal splines.” J. Res. Nat. Bur. Stand. 87 (4): 317–346. https://doi.org/10.6028/jres.087.021.
Tari, E., and O. Baykal. 2005. “A new transition curve with enhanced properties.” Can. J. Civ. Eng. 32 (5): 913–923. https://doi.org/10.1139/l05-051.
Vázquez-Méndez, M. E., and G. Casal. 2016. “The clothoid computation: A simple and efficient numerical algorithm.” J. Surv. Eng. 142 (3): 04016005. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000177.
Vázquez-Méndez, M. E., G. Casal, D. Santamarina, and A. Castro. 2018. “A 3D model for optimizing infrastructure costs in road design.” Comput.-Aided Civ. Infrastruct. Eng. 33 (May): 423–439. https://doi.org/10.1111/mice.12350.
Wang, L. Z., K. T. Miura, E. Nakamae, T. Yamamoto, and T. J. Wang. 2001. “An approximation approach of the clothoid curve defined in the interval and its offset by free-form curves.” Comput.-Aided Des. 33 (14): 1049–1058. https://doi.org/10.1016/S0010-4485(00)00142-1.
Information & Authors
Information
Published In
Copyright
©2019 American Society of Civil Engineers.
History
Received: Nov 12, 2018
Accepted: Jul 15, 2019
Published online: Oct 31, 2019
Published in print: Feb 1, 2020
Discussion open until: Mar 31, 2020
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.