The statistical analysis of aircraft trajectories within the framework of Functional Data Analysis (FDA) requires a set of preprocessing steps that are well known in the literature. We are interested in the problem of aligning trajectory data, which is common in many applications. For trajectory data, the presence of phase variations is generally inevitable because the studied flights are never simultaneous, have different durations, and are operated with significant variabilities across airlines. We propose a comparison of two registration methods: a landmark-based registration and an elastic registration. From a sample of flights, the objective is to construct an average altitude profile. This average profile should be as informative as possible in the sense that it should characterize the average altitude amplitude. A good registration procedure should therefore summarize altitude variations for similar flight phases. In the ideal scenario where flight phases are provided in the raw data, landmark registration is natural: it suffices to identify landmarks at phase changes. If this is not the case, flight phases can be segmented beforehand. Leveraging a more advanced conceptual framework based on differential geometry, we highlight that elastic registration produces a more relevant average altitude profile. Thanks to a well-chosen Riemannian metric, the registration enables the clear distinction of plateaus in the approach phase, even when information about flight phases is not explicitly used.