We introduce a new way to look at the combined lateral and longitudinal control problem for platooning vehicles by studying these problems separately. The lateral control problem is approached as a path following problem in the spatial domain: based on the path of the preceding vehicle we determine a path for the following vehicle which converges to the given path of its predecessor. In particular if the following vehicle happens to be on the path of its predecessor, the generated path of the follower equals the path of its predecessor. This approach not only overcomes the problem of corner cutting, but also achieves appropriate following behavior in case of large initial errors. As a by-product of solving the lateral control problem, we obtain a mapping from the path of the follower to the path of its predecessor. Using this mapping we can consider the longitudinal control problem as controlling two points on the same path towards a required inter-vehicle distance, which is comparable to CACC, i.e., the problem of controlling two points on a straight line towards a required inter-vehicle distance. We illustrate our approach by means of simulation.