Modelling of Joining Route Segments of Different Curvature

Władysław Koc, Katarzyna Palikowska


The paper presents a new general method of modelling route segments curvature using differential equations. The method enables joining of route segments of different curvature. Transitional curves of linear and nonlinear curvatures have been identified in the case of joining two circular arcs by S-shaped and C-oval transitions. The obtained S-shaped curves have been compared to the cubic C-Bezier curves and to the Pythagorean hodograph quantic Bezier curve using the Lateral Change of Acceleration diagram and the dynamic model. The analysis of dynamic properties has showed an advantage of the obtained transition curve of nonlinear curvature over Bezier curves.


curvature; Bezier curve; differential equations; dynamic analysis; highway; lateral change of acceleration; transition curve.

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Ahmad, A.; Ali, J. 2008. G3 Transition Curve between Two Straight Lines, in Proc. of the 5th International Conference on Computer Graphics, Imaging and Visualisation, 26–28 August 2008, 154–159.

Baykal, O. 1996. On Concept of Lateral Change of Acceleration, Journal of Surveying Engineering 122(3): 132–141.

Bosurgi, G.; Pellegrino, O.; Sollazzo, G. 2016. Using Genetic Algorithms for Optimizing the PPC in the Highway Horizontal Alignment Design, Journal of Computing in Civil Engineering 30(1).

Bosurgi, G.; D’Andrea, A. 2012. A Polynomial Parametric Curve (PPC-Curve) for the Design of Horizontal Geometry of Highways, Computer-Aided Civil and Infrastructure Engineering 27(4): 304–312.

Cai, H.; Wang, G. 2009. A New Method in Highway Route Design: Joining Circular Arcs by a Single C-Bezier Curve with Shape Parameter, Journal of Zhejiang University SCIENCE A 10(4): 562–569.

Habib, Z.; Sakai, M. 2007a. G2 Pythagorean Hodograph Quintic Transition between Two Circles with Shape Control, Computer-Aided Geometric Design 24(5): 252–266.

Habib, Z.; Sakai, M. 2007b. On PH Quantic Spirals Joining Two Circles with One Circle Inside the Other, Computer-Aided Design 39(2): 125–132.

Kobryń, A. 2014. New Solutions for General Transition Curves, Journal of Surveying Engineering 140(1): 12–21.

Kobryń, A. 2011. Polynomial Solutions of Transition Curves, Journal of Surveying Engineering 137(3): 71–80.

Koc, W. 2014. Analytical Method of Modelling the Geometric System of Communication Route, Mathematical Problems in Engineering, ID 679817, 13 p.

Koc, W.; Chrostowski, P. 2013. Computer-Aided Design of Railroad Horizontal Arc Areas in Adapting to Satellite Measurements, Journal of Transportation Engineering 140(3). ID 04013017.

Koc, W.; Palikowska, K. 2012. Dynamic Properties Evaluation of the Selected Methods of Joining Route Segments with Different Curvature, Technika Transportu Szynowego 9: 1785–1807 (Polish).

Koc, W.; Mieloszyk, E. 1998. Mathematical Modeling of Railway Track Geometrical Layouts, Archives of Civil Engineering 44(2): 183−198.

Li, X.; Li, M.; Wang, H.; Bu, J.; Chen, M. 2010. Simulation on Dynamic Behavior of Railway Transition Curves, ICCTP 2010 Integrated Transportation Systems: Green, Intelligent, Reliable 3349–3357.

Long, X.; Wei, Q.; Zheng, F. 2010. Dynamic Analysis of Railway Transition Curves, in Proc. of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 224(1): 1–14.

Mieloszyk, E.; Koc, W. 1991. General Dynamic Method for Determining Transition Curve Equations, Rail International - Schienen der Welt 10: 32–40.

Tari, E. 2003. The New Generation Transition Curves, ARI The Bulletin of the Istanbul Technical University 54(1): 34–41.

Ziatdinov, R.; Yoshida, N.; Kim, T. 2012a. Analytic Parametric Equations of Log-Aesthetic Curves in Terms of Incomplete Gamma functions, Computer Aided Geometric Design 29(2): 129–140.

Ziatdinov, R.; Yoshida N.; Kim, T. 2012b. Fitting G2 Multispiral Transition Curve Joining Two Straight Lines, Computer-Aided Design 44(6): 591–596.

DOI: 10.3846/bjrbe.2016.01


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