New Solutions to Optimum Vertical Curve Problem

Estimating curve parameters using observed data is of fundamental importance to surveyors. In 1999 Easa converted the $L_1$ estimation of parabolic curve parameters into a linear programming (LP) problem, which enabled a rapid solution by LINDO. However, this approach requires LP formulation and associated software that may not be easily inaccessible to all surveyors, while its efficiency declines when handling the nonlinear global problem with unknown start and end points $(x_1$ and $x_2)$ of the curve. Easa sought the global solution by repeating the LP procedure over various combinations of $x_1$ and $x_2$ manually, using a step size of 5 m for each variable. With this approach, one faces a trade-off between labor and accuracy. As an alternative, a convenient spreadsheet method suitable for $L_1\text{,}$ $L_2\text{,}$ or “mini-max” optimization is developed here that reduces the labor and formulation involved and requires no LP language. The new approach is further automated by Visual Basic for Applications programming to solve the global problem, allowing a fine step size of 0.1 m to be used. The vast data set generated leads to an accurate 3D picture that reveals the true behavior of the global solution, which cannot be captured when a step size of 5 m is used.