Defining topographic trajectories with a minimum longitudinal area

The problem of finding the trajectory with the minimum excavation volume between two given points of a terrain represented by a digital elevation model is analyzed. Two approaches based on the calculus of variations (CVs) and genetic algorithms (GAs) were used to solve this problem. Initially, a terrain with the shape of an axisymmetric monticule modelled by an exponential function, where the contour lines are concentric circumferences, was analyzed by means of CV procedures. A trajectory linking two points symmetrically located with respect to the centre of the monticule is sought. For this case, it has been possible to predict the trajectory shape as a function of the distance from the extreme points to the centre. It was found that the trajectory can be a straight line passing through the centre or it can have an elliptical, circular or parabolic shape. We also analyzed a case in which the extreme points of the trajectory are not symmetric with respect to the monticule centre. The trajectories with the minimum excavation volume are useful when locating roads, channels and rail networks, thus the use of GA to find them is of great interest to a wide field of engineering practice.