While the classical Schwarz method can only be used with overlap, optimized Schwarz methods can also be used without overlap, which can be an advantage when simulating heterogeneous problems, problems with jumping coefficients, or also for independent mesh generation per subdomain. The analysis of nonoverlapping optimized Schwarz methods has so far been restricted to the case of straight interfaces, even though the method has been successfully used with curved interfaces. We close this gap by presenting a rigorous analysis of optimized Schwarz methods for circular domain decompositions. We derive optimized zeroth and second order transmission conditions for a model elliptic operator in two dimensions, and show why the straight interface analysis results, when properly scaled to include the curvature, are also successful for curved interfaces. Our analysis thus complements earlier asymptotic results by Lui for curved interfaces, where the influence of the curvature remained unknown. We illustrate our results with numerical experiments.