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Geometry and the Explanation Problem


The subject of geometry has not been ``computerized'' in the same way as algebra, in that there are no Computer Geometry Systems analogous to existing Computer Algebra Systems such as Axiom, Maple, Weyl, and Mathematica. The closest thing to Computer Geometry Systems are commercial Computer Aided Design (CAD) packages such as AutoCad.

This emphasis on CAD (and on producing pictures of objects) has led to another problem with existing geometric software packages: these packages typically provide a severely limited set of possible geometric representations. The provided representations are useful for CAD, but are too restrictive for more general mathematical applications. Clearly, no single representation can support all possible geometric objects, but consider one popular (abstract) way to represent such objects: boundary representation. In a boundary representation, the geometric object is stratified; zero-dimensional elements of its boundary are described, then these are linked to one-dimensional boundary elements, and so on, for as many dimensions as needed. There are no widely-supported and fully general realizations of boundary representations in current mathematical software. Instead, existing software typically imposes limitations such as geometric objects can be three-dimensional only, coordinates are required to be represented in floating-point (as opposed to allowing additional numeric systems such as rational or algebraic numbers), and only certain kinds of parametric surfaces are allowed.

nuprl project
Tue Nov 21 08:50:14 EST 1995