In the "Budded" system (Fig. ), while at small lagtimes the center of the bud (r = 0) exhibits the largest conventional MSD values, this is completely obscured by the mixing with molecules originating from regions of lower apparent diffusion coefficient (the sides of the bud). Such a qualitative change in the apparent mobility of various regions has direct implications on the interpretation of experimental results on curved membranes, as it renders When comparing the conventional and the gMSD results, it becomes clear that using the projected values manifests as an artificial slowdown on surfaces of both mean and Gaussian curvature, with a magnitude roughly proportional to the curvature. When geodesic displacements are used, all MSD curves shift to higher values and become smoother as a function of the spatial coordinates. In addition, similarly to the conventional MSD, molecules originating from regions of different curvature gradually mix as the lagtime increases, producing uniform MSD profiles at large enough lagtimes. To better quantify these effects, we computed the diffusion coefficient distributions using the conventional MSD calculation method and the geodesic-based gMSD approach presented here. Furthermore, we also decomposed the latter based on local curvature (see Supporting Information for details). The diffusion coefficients at the individual mesh points were calculated at the lag time of 53 ns, instead of separately applying linear regression to the MSD curves. It must be noted, that computing a diffusion coefficient distribution on surface mesh points is not strictly equivalent to the perparticle distribution, as less frequently occupied mesh points should have a lower statistical weight. However, the almost uniform density of lipid centers of mass (see Fig. and) indicates that the distinction should be only minor. Indeed, the simple division using one lagtime per mesh point MSD curve provided a very similar value to linear fits to MSD data calculated for lipids, validating our approach. The distributions of the diffusion coefficients in different membrane environments, shown in Fig. , confirm that assuming 2D movement significantly underestimates the diffusion coefficient of molecules moving on curved surfaces (compare Projected and Geodesic). Additionally, while the Flat region of the "Budded" system corresponds to the "apparently fastest" domain in the Projected distribution (both are around 6 • 10 -7 cm 2 /s), the Flat part of the "Wave" is conclusively faster than the projection. This is in agreement with the orientation of the planar membrane regions in the two systems (cf. Fig. ).