Maps are one of the most efficient ways to communicate information. They help people to make decisions in navigation, spatial planning, or risk and disaster management. Maps also communicate geopolitical information, they give a spatial dimension to rhetoric arguments, and generally aid the process of public opinion and consensus building. Effective maps immediately convey their message and hence are as simple as possible. Schematization creates a simplified and compact representation of the original data and reduces the visual complexity of maps. Linear features, such as roads and rivers, and the boundaries of regions are often drawn using only a few straight line segments in few different directions, or they are approximated by a few simple curves. Traditionally schematized maps make extensive use of curves. Curves have greater expressive power than line segments, several line segments can often be replaced by a single arc of a low degree curve. Curves also make it easier for users to interpret maps. However, automated methods for schematization are mostly restricted to straight lines. Decision makers and the greater public benefit from high quality on-demand map production, which necessarily has to be fully automated. Hence we propose to develop algorithmic methods for automated curved schematization. We aim to study curved schematization for region outlines, subdivisions, and networks. We will establish cartographic quality criteria and develop efficient algorithms that take these into account. Formalizing these criteria and testing their validity will be an iterative process executed simultaneously with algorithm development and experimental evaluation.