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## A Two Dimensional Derivative of Boundary Representation

The second approach which I took to two dimensional qualitative geometry was developed from boundary representation methods rather than from constructive solid geometry methods. The reason for this was that the most important aspects of the ASSF representation in reasoning about motion seemed to be boundary related rather than axis related, as I explain later.

Solid modelling using boundary representation requires a larger amount of information to describe basic three dimensional shape than the combination of constructive solid geometry and generalised cones does, but it can also describe a wider range of objects. This is because all surfaces of the object are described explicitly, whereas in a CSG description they are implicit in the combination of primitives, and in the shape sweeping methods used for describing primitives.

Explicit boundary description provides important advantages where the representation is used for reasoning about interaction between objects, rather than just properties of a single object. This is because objects only contact other objects on their boundaries, so a description of the boundary must be available to the reasoning system, whether it is given explicitly, or determined by computation from a constructed solid description.

The generalised cones method requires that a method of representing two dimensional shape be used to describe the cross section for ``sweeping'' operations, but a boundary representation must describe the two dimensional shape of every face in the three dimensional object. This has the advantage that individual features are separately described in local two dimensional contexts for the whole object, whereas for CSG, features in planes other than the cross section must be inferred from the sweeping function.

An object boundary can be qualitatively described in two dimensions by identifying sections that are qualitatively homogeneous, then describing the relationships between those sections. If the homogeneous sections were all straight lines, the resulting description would be a polygon. For more generalised shape description, the sections can also be curves or wiggles, and the description becomes an extended polygon.

The qualitative representation which I derived from three dimensional boundary representation describes shape in exactly this way - as a collection of qualitatively different segments arranged to form an extended polygon. For brevity, I will refer to this extended polygon boundary method as EPB.

Next: Qualitative Two Dimensional Shape Up: 2D Qualitative Geometry from Previous: A Two Dimensional Derivative
Alan Blackwell
2000-11-17