This paper proposes a methodology for determining the shape and ultimately the functionality of objects from intensity images; 2D analytic functions are used to track 3D features during known camera motions. Three analytic functions are proposed that describe the relationship between pairs of points that are either stationary or moving depending on whether the points are on occluding boundaries or otherwise. Many of the problems of correspondence are reduced by using foveation, known camera motion, and active vision methods. The three analytic functions are shown to enable hypothesis refinement of the functionality of a number of 3D objects without full 3D information about the shape.