This paper is about using interval computations in location, simplification, and root-finding for multivariate implicit functions that are used as shape primitives in a set-theoretic (that is, a CSG) geometric modeller. Three problems are discussed, and solutions to them presented: the location and simplification of the surfaces of semialgebraic sets (surfaces involving some transcendental functions are dealt with as well); the generalization of Newton-Raphson using intervals; and interval ray-tracing. Examples are presented for both conventional three-dimensional geometric models and for CSG models in higher dimensions representing configuration-space maps for moving and colliding three-dimensional objects.

Djinn is an API for solid modelling which is defined in the language of point-sets and is thus independent of any particular modelling data-structure (e.g. boundary representations or CSG trees). In designing this API, several significant and interesting questions have arisen, including: the basic feasibility of the approach; deciding how to support cellular models; providing facilities for navigation without traditional data-structures; addressing the problems of 'tweaks' and blends; ensuring that labels are preserved under geometric operations; permitting regions in which changes to a model have taken place to be identified; coping with approximate models; deciding whether to support variational models; keeping sweeps and transforms within a reasonable domain; and providing facilities to handle the many types of parametric surface. These questions are not all fully answered, but progress continues towards a published interface and C++ language binding.

This paper presents a new solid modelling system capable of representing three dimensional objects consisting of both simple and complicated surfaces and, equally importantly, blend surfaces between them. Implicit polynomial inequalities and set theoretic techniques are used to specify the models. The modelling system can generate shaded pictures and NC machining instructions for cutting complicated objects automatically.

An efficient algorithm is proposed for computing the Dirichlet tessellation and Delaunay triangulation in a k dimensional Euclidean space (k≥2). The algorithm is designed in a way that should allow it to be extended to some of the simpler non-Euclidean metric spaces as well. The algorithm has been implemented in ISO FORTRAN by the author and execution time and stereoscopic pictures of the tessellation and triangulation are presented at the end of this paper.

A multiprocessor architecture for the solution of spatial and graphical problems by recursive subdivision is proposed. The trees that result from such divisions are mapped on to a closed graph, which is then considered as a network of processors. Simulations of these configurations have been performed with encouraging results. Proposals are made for the implementation of such machines.