The project ”GAIA II - ”Intersection algorithms for geometry based IT-applications using approximate algebraic methods” combines knowledge from Computer Aided Geometric Design (CAGD) and classical algebraic geometry to improve intersection algorithms for Computer Aided Design (CAD) types system. The focus within the project is on: Exact and approximate implicitization, Classification and identification of singularities, Recursive subdivision based intersection algorithms, Industrial testing At a first glance to calculate the intersection between curves or surfaces can seem simple. This is true for the intersection of e.g. two straight lines when they intersect transversally. For the intersection of two straight lines closed expressions for the intersection exist. However, if the lines are near parallel care has to be taken to implement a stable solution using floating point arithmetic. When we intersect two bi-cubic parametric surfaces the problem can be reduced to the zero set of a polynomial equation f(s,t)=0 of degree 54 in both variables, which by itself is a challenging problem. In industrial systems (e.g. CAD) double precision floating point arithmetic is used, thus introducing rounding errors. In CAD system there are tolerances defining when two points are to be regarded as the same point. This has also to be taken into consideration in CAD-related intersection algorithms. Low quality of intersection algorithms in CAD-systems imposes high costs on the product creation process in industry.

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