In haptic exploration, when running a fingertip along a surface, the control system may attempt to anticipate upcoming changes in curvature in order to maintain a consistent level of contact force. Such predictive mechanisms are well known in the visual system, but have yet to be studied in the somatosensory system. Thus the present experiment was designed to reveal human capabilities for different types of haptic prediction. A robot arm with a large 3D workspace was attached to the index fingertip and was programmed to produce virtual surfaces with curvatures that varied within and across trials. With eyes closed, subjects moved the fingertip around elliptical hoops with flattened regions or Limaçon shapes, where the curvature varied continuously. Subjects anticipated the corner of the flattened region rather poorly, but for the Limaçon shapes they varied finger speed with upcoming curvature according to the two-thirds power law. Furthermore, although the Limaçon shapes were randomly presented in various 3D orientations, modulation of contact force also indicated good anticipation of upcoming changes in curvature. The results demonstrate that it is difficult to haptically anticipate the spatial location of an abrupt change in curvature, but smooth changes in curvature may be facilitated by anticipatory predictions.