Axisymmetric shock reflection occurs in a wide range of aerospace applications such as supersonic/hypersonic intakes and nozzles. An analytical model to predict the curved incident shock generated through ring-shaped intakes with streamwise curvature has been derived in this study by extending a preceding model proposed by Ren et al. for ring intakes without curvature, assuming the streamline curvature behind the incident shock to be the same as that of the ring curvature. It has been validated by performing numerical simulations with different ring shapes and inflow conditions, focussing on the region of the incident shock that is not affected by the expansion waves generated from the trailing edge of the ring. The extended model has been found to be more accurate than that of the original model for all cases, while some errors have persisted even with the extended model. The errors have been attributed to inadequate modelling of the streamline curvature behind the incident shock, which varies toward the symmetry axis.