Vertical vs. Horizontal Fractal Dimensions of Roads in Relation to Relief Characteristics

Abstract

This paper investigated the surface length of roads from both horizontal and vertical perspectives using the theory of fractal dimension of surfaces and curves. Three progressive experiments were conducted. The first demonstrated the magnitude of the differences between the planar road length and the DTM-derived surface road length and assessed its correlation with the DTM-calculated road slope. The second investigated the road distance complexity through the fractal dimension in both planar and vertical dimensions. The third related the vertical with the horizontal fractal dimension of roads across a range of distinct physiographic regions. The study contributed theoretically by linking the planimetric complexity to vertical complexity, with clear applications for advanced transportation studies and network analyses. The core methodology used geographic information systems (GIS) to integrate a high resolution (1 × 1 m) digital terrain model (DTM) with a road network layer. A novel concept, the vertical fractal dimension of roads was introduced. Both the vertical and horizontal fractal dimensions of the roads were calculated using the box-counting methodology. We conducted an investigation into the relationship between the two fractal dimensions using fourteen study areas within four distinct physiographic regions across Slovenia. We found that the average slope of a three-dimensional (3D) road was directly related to the length difference between 3D and two-dimensional (2D) roads. The calculated values for the vertical fractal dimension in the study areas were only slightly above 1, while the maximum horizontal fractal dimension of 1.1837 reflected the more sinuous properties of the road in plan. Variations in the vertical and horizontal fractal dimensions of the roads varied between the different physiographic regions.