Non-contact atomic force microscopy is used to measure the 3D force field on a dense-packed Cu(111) surface. An unexpected image contrast reversal is observed as the tip is moved towards the surface, with atoms appearing first as bright spots, whereas hollow and bridge sites turn bright at smaller tip-sample distances. Computer modeling is used to elucidate the nature of the image contrast. We find that the contrast reversal is essentially a geometrical effect, which, unlike in gold, is observable in Cu due to an unusually large stability of the tip-sample junction over large distances.