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Several fields of mathematics are relevant to computer aided design and other software systems involving solid object geometry, topology, differential and algebraic geometry being particularly important. This paper discusses some of this mathematics in order to provide a theoretical foundation for geometric modelling kernels that support non-manifold objects with an internal cellular structure and subsets of different dimensions.

The paper shows relationships between relevant concepts from topology, differential geometry and computer aided geometric design that are not widely known in the CAD community. It also discusses semialgebraic, semianalytic and subanalytic sets as candidates for object representation. Stratifications of such sets are proposed for an object's cellular structure and new stratification concepts are introduced to support candidate applications.

Current shape models are targeted at visual presentations for display and design. They lack the validity in their shape properties such as topological-, geometrical- and visual- equivalence, and even continuity. Cellular modeling is a new computational modeling that provides a computationally valid shape model. It also provides a foundation to share shapes among varied applications for extensive reuse. The implementation of cellular modeling via cell attachment tables complies with the standard relational data model. Examples are shown to demonstrate the value of cellular modeling in comparison with the existing typical shape models such as wire frame models, boundary models and solid models. Design and implementation of the cellular modeling examples using cell attachment instance tables are presented.