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Show 2 Theoretical and algorithmic advances in the mathimathical methods have included some basic spline results such as Developing theory and practical algorithms, based on discrete splines, for degree raising of general spline curves. These results had immediate applications in the research performed on new 3-D modelling techniques for computer aided design. Initial investigations of multivariate spline theory with our Oslo collegue Professor Tom Lyche. In that context we have introduced "discrete box splines", which serve as a theoretical framework for refinement theory of continuous box splines. The proofs and algorithms developed investigating the relationships between discrete and continuous box splines open the door for investigation of other properties of continuous box splines. Algorithms for performing boolean operations on polyhedra, that is union, intersection, and difference have been developed. While the theory is relatively straightforward, the implementation in floating point brings this problem into the domain of fuzzy sets. Exact arithmetic was tried but found to perform unacceptably slow. The study of methods for using this theory in the design arena involved * Research into modelling the standard "primitives" of solid modelling using the representation proposed in this contract. * Developing spline defined "rounded edge primitives". * Investigating new operators which might act as the "verbs" of computer aided mechanical design. A modelling project for a freeform object, a helicopter was initiated as a study in tools and techniques. Thus, investigations into many interesting problems deriving from the applications of discrete splines to the problems of computer aided geometric design. New theory and algorithms have been developed and support mechanisms based on the Oslo algorithm have been incorporated in order to help use the geometry model directly to calculate many geometric attributes needed for the design process including surface rendering and intersections. |
