Optimum Vertical Curves for Highway Profiles

Information on the vertical alignment of existing old highways is often not available. To evaluate the highway alignment, it is necessary to fit vertical curves (parabolic curves and associated tangents) to the observed profile data. This paper presents a linear optimization model that determines the best parabolic curve/tangents that fit a given profile. The decision variables of the model are the grades of the first and second tangents and the location of the start and end of the parabolic curve. The objective function of the model minimizes the sum of the absolute deviations between the observed profile and the vertical curve. The model includes a constraint to make the point of vertical intersection lie in the middle of the curve, thus ensuring that the estimated grades are indeed tangents to the parabolic curve. Extensions of the model to accommodate another objective and multiple vertical curves are presented. The application of the model was illustrated using a numerical example involving a crest vertical curve. The proposed model can be easily solved using optimization software that are readily available in practice.