Finding the optimal route through the terrain is a problem often encountered in practice, especially in the construction industry. Planning a new transportation route presents several challenges - one of them is an accurate representation of the terrain which is usually determined through multiple height points. The common approach uses linear spline interpolants to find a surface which interpolates the given points. The disadvantage of doing so is that interpolants aren’t smooth and don’t meet the standards required. To address this, we will first examine the theory of approximation and interpolation of smooth surfaces. Secondly, we will look into the criterion of optimality and the shortest path algorithm. Finally, we will attempt to calculate the optimal route of the train track between Divača and Koper using minimised travel distance. From a set of given points we will generate a terrain using linear spline interpolation over which we will attempt to draw the shortest path. To achieve this, the maximum allowed steepness of the terrain and the maximum allowed angle at a turn point will be used. We will then compare our results to the existing track and the current plan for the new one.
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