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For a four-point mission like Cluster, the differences in position and timewhen the satellites detect the magnetopause or any other discontinuity, canbe used to infer the discontinuity local orientation, thickness and motion.This timing technique, commonly assuming a planar geometry, offers anindependent check for various single-spacecraft techniques.  In the present paper we propose an extension of the timing method, capable ofdetermining in a self-consistent way the macroscopic parameters of atwo-dimensional, non-planar discontinuity. Such a configuration can beproduced by a local bulge or indentation in the magnetopause, or by a largeamplitude wave traveling on this surface, and is recognized in Cluster datawhen the single spacecraft techniques provide different individual normalscontained roughly in the same plane. The model we adopted for themagnetopause assumes a layer of constant thickness of either cylindrical orparabolic shape, which has one or two degrees of freedom for the motion inthe plane of the individual normals. The method was further improved byincorporating in a self-consistent way the requirement of minimum magneticfield variance along the magnetopause normal. An additional assumption,required in a previously proposed non-planar technique, i.e. that thenon-planarity has negligible effects on the minimum variance analysis, isthus avoided.  We applied the method to a magnetopause transition for which the variousplanar techniques provided inconsistent results. By contrast, the solutionsobtained from the different implementations of the new 2-D method wereconsistent and stable, indicating a convex shape for the magnetopause. Thesesolutions perform better than the planar solutions from the normal magneticfield variance perspective. The magnetopause dynamics and the presence of anon-zero normal magnetic field component in the analyzed event are discussed

A new technique for determining orientation and motion of a 2-D, non-planar magnetopause