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The fields transmitted from a line source through a two-dimensional curved dielectric layer of variable thickness are constructed by geometric-optical ray tracing that accounts for multiple reflections on the concave side, where the source is located, as well as for reflections between the layer boundaries. Moreover, internally trapped rays, excited by evanescent tunneling, are included when source and observer are near the layer boundaries but laterally displaced along it. By applying Poisson summation to the ray series, the multiple reflected contributions, for weakly tapered configurations, can be summed to yield trapped and leaky local modes guided along the layer, as well as a "collective" ray field that incorporates a plane layer transmission coefficient, with curvature and slope corrections, instead of the conventional coefficients for individual boundaries. Detailed calculations are performed for the special cases of a circularly curved layer of constant thickness and a tapered layer with nonparallel plane boundaries. For the former, the various ray-optically derived solutions agree completely with those obtained from a rigorous analysis. For the latter nonseparable configuration, no rigorous solution is available. With direct summation of conventional ray fields taken as a reference, extensive numerical results demonstrate the economies effected by the collective ray formulation and the importance of including the curvature or slope corrections in the equivalent slab transmission coefficients.