Geodesic Path Computation on Connected Trimmed Surfaces

Creeping waves considerably contribute to the total high-frequency electromagnetic field in the shadow region. These waves propagate along surface geodesic paths. To trace them, geodesic differential equations have to be solved, which require special care in the case of connected trimmed surfaces. In this article, an algorithm is proposed to compute geodesic path transition across connected trimmed surfaces. The formulation includes efficient boundary recognition as well as accurately initializing geodesic equations on the next trimmed surface. For boundary recognition, inscribed or peripheral polygons are constructed to estimate the boundary and to effectively initialize numerical methods of quadratic convergence to determine the exact exit point. To get the initial point and initial direction on the parametric domain of the next trimmed surface, two optimization problems are efficiently solved. The proposed algorithm is applicable even if first/second-order surface discontinuity is encountered during the transition. The proposed method is validated on canonical surfaces and nonuniform rational B-spline (NURBS) surfaces for various trimming curves. Accurate computation of geodesic paths is observed. Furthermore, the far-field radiation pattern of a monopole antenna mounted on a sphere, a cylinder, and an ellipsoid is obtained and confirmed by numerical electromagnetic software.