A formal specification of geometric refinements

The geometric refinements of 2D and 3D subdivisions are basic operations in geometric programming. They consist in partitioning their cells, i.e. their volumes, faces and edges, until no intersection exists between them and in achieving their topological restructuring. These crucial operations deserve formal and precise definitions. We present, in this paper, new methods to formalize their design. Starting from a mathematical definition of the topological models and refinements, we present an algebraic specification of the operators needed to handle them. Then, we give a high-level and complete description of the refinement processes thanks to the use of rewrite systems. This approach allows us to exhibit integrity constraints for the handled objects and to focus on the conceptual and logical aspects of the refinement, avoiding tedious details of implementation. Finally, we show how the systems are enhanced to reflect choices of implementations and algorithmic improvements