Algebraic Surface Derivatives for Rendering Virtual Contact Force

Previous augmented reality displays have rendered crisp virtual walls and virtual objects with finger-surface reactions through haptics feedback devices. The widely used stiffness (or impedance) methods penalize a user’s intrusion into a virtual surface with a force response dependent on the amount of penetration along the contact normal at the virtual proxy location. We present an alternative approach for computing the surface impact response that projects Lagrange multipliers associated with a unilateral surface constraint onto a multivariate surface constraint Jacobian for the case where the two objects (finger and model) do not stick together. Advantages of our method are that the surface Jacobian is already being computed from an exact NURBS-NURBS collision update algorithm previously developed by the authors, and that the force response is accurate to numerical precision; no polygonal or numerical approximations are employed.