AbstractVertical curve fitting is important both for road safety management and railway maintenance. Different parameterizations were proposed to specify a vertical curve satisfying the constraint of continuity between its components. The vertical curve-fitting model was developed to explore the effects of different parameter sets on the fitting performance for a vertical curve. The steepest descent (SD), Gauss-Newton (GN), and Levenberg-Marquardt (LM) algorithms were used to search for the optimum solution. Experiments showed that the different parameterizations have different effects on the performance of the three algorithms. For one parameter set, the three algorithms converged to different solutions. For the other, the three algorithms converged to the same optimal solution. The two parameterizations responded differently to initial values, which were illustrated visually. From six different initial values, the LM algorithm converged to different solutions for one parameterization, but to the same optimum for the other. The condition number (CN), an index for evaluating the extent of ill-posed problems as well as correlations between parameter pairs, was examined to interpret the reasons of one parameterization outperformed the other in both robustness and effectiveness. Results also showed that the LM algorithm outperformed the GN algorithm in robustness and outperformed the SD algorithm in efficiency.

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