LEAST STRATIFICATIONS AND CELL-STRUCTURED OBJECTS IN GEOMETRIC MODELLING

Abstract

Results from the study of O-minimal structures are used to formalise fundamental objects and operations for geometric modelling kernels. O-minimal structures provide a more general setting than the traditional semialgebraic sets; they allow semianalytic sets such as screw-threads to be modelled accurately. O-minimality constrains the class of sets to ensure the retention of the key finitary properties necessary for many geometric operations. The formalism is independent of internal representations and implementations, in the spirit of the mixed-dimension cellular objects of the Djinn API (Application Programming Interface) [2].Topological stratification is used to provide an object with a cellular structure consisting of a finite set of piecewise-smooth manifold cells. The main contribution of the paper is an existence proof for least stratifications (of sets in O-minimal structures). Set-like combination of two cell-structured objects requires a common partition of space from which a cell-structured result can be obtained. The existence of least stratifications facilitates the definition of combination operators that provide unique results.