The Role of Gaussian and Mean Curvature in 3D Highway Geometric Design and Safety

This study investigates the use of three-dimensional (3D) roadway surface-based geometric indicators in traffic crash analysis, with the objective of evaluating their potential to represent the combined effects of highway alignment features more effectively than traditional two-dimensional (2D) indicators. The roadway surface is modeled as a continuous 3D B-spline surface, from which surface-based geometric metrics derived from differential geometry—specifically Gaussian curvature and mean curvature—are calculated. The roadway is segmented into fixed-length surface patches, and crashes are spatially allocated to these patches using a point-in-polygon approach. Patch-level crash frequencies are analyzed using negative binomial regression models, with traffic exposure accounted for through annual average daily traffic (AADT). The results demonstrate that surface-based 3D curvature metrics are statistically significant explanatory variables in crash frequency modeling and are capable of capturing geometric interactions that are not explicitly represented by conventional 2D alignment measures. The proposed framework provides a proof-of-concept for incorporating 3D roadway geometry into highway safety analysis and offers a foundation for future development of integrated, surface-based crash prediction models.