Haptic, or force, interaction in virtual environments can support the development of physical manipulation skills if users feel realistic forces in response to their motion in the virtual environment. The realism of the forces felt by users depends on:​ (i) how accurately the virtual environment simulates a real life situation such as surgery;​ and (ii) how faithfully the haptic controller renders the simulated interactions to users. Accurate simulations of real life situations such as surgery run at variable frequencies of the order of 20-100 Hz. However, the haptic controller needs updated stiffness and direction of contact at 1000 Hz to faithfully convey the shape and hardness of the virtual objects to the user. This thesis proposes to bridge the gap between the required fast haptic control rate and the slower virtual environment updates through a passive local model of interaction. This model comprises an approximation of the shape and stiffness of the virtual world in the area near the point of interaction. It also monitors its exchange of energy with the user to ensure its own passivity and thus, the stability of the haptic system. Lastly, the local model eliminates the spurious discontinuities that arise in contact direction at model updates by interpolating the contact normal before rendering it to the user.