AIMS: We want to understand the emission mechanism of radio emission from air showers to determine the origin of high-energy cosmic rays. Therefore, we study the geometry of the air shower radio emission measured with LOPES and search for systematic effects between the direction determined on the radio signal and the direction provided by the particle detector array KASCADE. METHODS: We produce 4D radio images on time-scales of nanoseconds using digital beam-forming. Each pixel of the image is calculated for three spatial dimensions and as a function of time. The third spatial dimension is obtained by calculating the beam focus for a range of curvature radii fitted to the signal wave front. We search this multi-dimensional parameter space for the direction of maximum coherence of the air shower radio signal and compare it to the direction provided by KASCADE. RESULTS: The maximum radio emission of air showers is obtained for curvature radii being larger than 3 km. We find that the direction of the emission maximum can change when optimizing the curvature radius. Furthermore, we find a tentative increase of the curvature radius to lower elevations, where the air showers pass through a larger atmospheric depth. CONCLUSIONS: We conclude that the angular resolution of LOPES is sufficient to determine the direction which maximizes the observed electric field amplitude. However, the statistical uncertainty of the directions is not determined by the resolution of LOPES, but by the uncertainty of the curvature radius. We do not find any systematic deviation between the directions determined from the radio signal and from the detected particles. This result places a strong supportive argument for the use of the radio technique to study the origin of high-energy cosmic rays.