Some methods of symbolic reasoning about the interaction between shapes and their function are summarized; the analysis is restricted to the interaction between pairs of two-dimensional objects, each with only one degree of freedom. The intent is to drive the reasoning from first principles based on the qualitative representation of kinematic function.  Faltings  proposes a computational theory based on specifying dependencies between shape features in terms of metric parameters, which in turn determine kinematic behavior; the objective is to investigate the interactions of shapes considering the interdependence of the metric parameters. Reasoning is done about one parameter at a time.

With the dimensions of the shapes given in terms of a set of parameters, kinematic function can be described by predicates defined in terms of these parameters and represented in a metric diagram. The metric diagram consists of a symbolic representation of the boundaries of each object defining a symbol for each metric dimension; a representation of knowledge about the metric dimensions, which could be a list of numerical values, a set of intervals, or a measurement procedure that determines a value when needed; and additional annotations such as the freedom of motion of an object or the fact that a shape is periodic. The two characteristics studied are the points of contact between parts and the way forces and motions are transmitted by the contact points.

The methods described could be applied to spatial reasoning in mechanical design, troubleshooting, and reasoning under uncertainty. In the words of the author, however, “at this time we do not have a characterization of the class of problems to which the methods of this paper could be applied.”

This paper is a report of an ongoing investigation. It is based on earlier work done by the author in collaboration with others on methods for the qualitative representation of kinematic function. Future work is promised on new methods that could be generalized to higher dimensions and to more degrees of freedom.

The paper will be of interest to researchers in qualitative kinematics and to graduate students. A full understanding of its contents will require access to the earlier work on which it is based.