Dimension-Independent Convex-Cell Based HPC Skeletons and Product

This article describes two new operations implemented within the design language, and discusses the advantages deriving from the definition and implementation of modeling operators within the HPC (Hierarchical Polyhedral Complex) representation scheme, which is used in the geometric nucleus of the language. The HPC representation scheme describes the geometric shapes as hierarchical collections of polyhedra, where each elementary polyhedron is decomposed into a set of convex cells. Each convex cell is in turn represented as a collection of either vertex vectors or facet covectors. In such an approach an object is represented as a hierarchical assembly. The HPC representation allows to take advantage from the locality properties of many operators, which are implemented by traversing the object structure and by applying the operator to each component. The structured object resulting from the application of an operator will reflect the meaning of the operator (see e.g. the product of polyhedra), so that the "structure" of the generated model is easily associated to the design process. A decompositive representation in convex cells is used in the representation of each elementary polyhedron. Along with each cell a reach symbolic description is maintained, and thus the use of numerical information is kept to a minimum. In this way the emphasis is shifted from numeric to symbolic information, with the first being considered less entrustable than the second, thereby increasing the robustness of the algorithms.