Function representation in geometric modeling: concepts, implementation and applications

Geometric modelling using continuous real functions of several variables is discussed. Modelling concepts include sets of objects, operations and relations. An object is a closed point set of n-dimensional Euclidean space with a defining inequality f x x xn (,,..., ) 1 2 0 °Ÿ¸. Transformations of a defining function are described for the set-theoretic operations, blending, offsetting, bijective mapping, projection, Cartesian product and metamorphosis. Inclusion, point membership and intersection relations are described. In the implemented interactive modelling system, we use highlevel geometric language that provides extendibility of the modelling system by input symbolic descriptions of primitives, operations and predicates. This approach supports combinations of representational styles, including constructive geometry, sweeping, soft objects, voxel-based objects, deformable and other animated objects. Application examples of aesthetic design, collisions simulation, NC machining, range data processing, and 3D texture generation are given