This paper addresses the problem of dexterous manipulation of objects using an articulated hand. We describe some geometric reasoning techniques that can be applied to solve active compliance tasks. In this context, fine motion strategies are derived from the positions and forces between the grasped object and the fingers of the hand. Our approach is based on the minimum power principle of the grasping system, using the force/position information to determine a sequence of well coordinated manipulation actions (fixed-point, rolling and sliding). Different manipulation situations are illustrated using a semi-spherical model of the fingertips. To test the techniques described in this paper, we present an example showing an articulated hand changing the orientation of a polyhedral object while respecting the stability constraints of the grasp.