An object model based on the principle of duality where a plane has a point as a dual and a point has a plane as a dual is presented. Using this representation we present methods for performing intersection detection and face overlap checks to determine if a parallel jaw gripper is in a position to execute a valid grasp. Valid grasps are ones physically realizable and can lift an object when it is assumed that the gripping forces and the friction coefficients are large enough to keep the object from slipping. The objects considered are polyhedra and valid grip checks for both convex and concave objects are developed. A generate and test method for synthesizing valid grasp positions is also presented.