Reverse GPS meter transformation

We are trying to anonymize GPS lat/long coordinates for one of our clients. The GPS coordinates represent trajectories sampled with a device at 0.5Hz frequency. We converted each trajectory to meters using a 3rd party library and then made all quantities relative to the origin or the trajectory(so first coordinates is (0, 0)). We also rotated the trajectory a random angle.

The client suggested that this is not really anonymous because the long->meter transformation depends on the origin point and this could be reversed. Our goal is to mask the original coordinates while keeping the shape of the trajectory. In our opinion there is no way our transformations could be reversed, especially since they involve randomness.

Is this really possible? Can you please post some reference paper?