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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

Tue Nov 21 08:50:14 EST 1995