The Frenet frame beyond classical differential geometry: Application to cartographic generalization of roads

Abstract

The reconstruction of curves from their curvature is a problem not very well-solved in cartographic generalization of roads. Frenet’s formulas theoretically solve this problem. The result shows that there exists a curve (up to rigid motion) with prefixed curvature function and torsion. By the use of the curvature and torsion functions, in closed forms, it is possible to obtain many beautiful examples in connection with this classical construction. In the case of plane curves, the torsion is equal to zero and the curve is characterized by the curvature. Nevertheless, if our starting point is a discrete set of arc length curvature experimental measures, the application of the Frenet frame is not straightforward. This study aims to describe how to adapt the classical Frenet frame to more realistic contexts derived from practical requirements. Our approach is applied to numerical measures derived from roads defined by Spanish Cartography, more precisely, by the Mapa Topográfico Nacional 1:25.000 (MTN25). The method described will be used to obtain generalizations of the original road by performing a standard wavelet decomposition on the curvature before the application of the Frenet frame for plane curves.