Tactile robots can perform complex interaction skills, e.g., polishing. Such robots should therefore be designed to be adaptive to environmental uncertainties such as changing geometry and contact-loss. To address this, we propose a tactile exploration technique to observe the local curvatures of the physical constraints such as corners, edges, etc. for updating predefined tactile skill policies accordingly. First, we develop a unified force-impedance control approach in which the force controller significantly improves the geometry following performance due to the ensured contact. Second, we use the proposed controller to autonomously investigate the unknown environment via the local curvature observer, designed to be a dynamic process. Finally, the exploration performance of the proposed controller is demonstrated by using a polishing skill on an unknown 3D surface, where the robot is observed to autonomously investigate the unknown surface from top to bottom along the edges and corners.