The peripheral endoplasmic reticulum (ER) forms different morphologies composed of tubules and sheets. Proteins such as the reticulons shape the ER by stabilizing the high membrane curvature in cross-sections of tubules and sheet edges. Here, we show that membrane curvature along the edge lines is also critical for ER shaping. We describe a theoretical model that explains virtually all observed ER morphologies. The model is based on two types of curvature-stabilizing proteins that generate either straight or negatively curved edge lines (R- and S-type proteins). Dependent on the concentrations of R- and S-type proteins, membrane morphologies can be generated that consist of tubules, sheets, sheet fenestrations, and sheet stacks with helicoidal connections. We propose that reticulons 4a/b are representatives of R-type proteins that favor tubules and outer edges of sheets. Lunapark is an example of S-type proteins that promote junctions between tubules and sheets. In a tubular ER network, lunapark stabilizes three-way junctions, i.e., small triangular sheets with concave edges. The model agrees with experimental observations and explains how curvature-stabilizing proteins determine ER morphology.