To better handle singular and near singular intersection configurations in surface intersections, the use of local approximate algebraic surfaces in CAD intersection algorithm development is an up-and-coming approach. To approximate larger regions of a CAD-model algebraically, using multivariate algebraic spline surfaces, will keep the polynomial degree lower, and allow for flexible approximations. The idea is to train a candidate from algebraic geometry in the concepts and approaches used in CAGD to establish a basis for the use of multivariate algebraic splines in CAD. To bring real algebraic geometry and spline surface representation in CAD closer together, we want to develop and understand certain aspects of the "classical" real (semi-) algebraic geometry so that it can extend to the theory of multivariate algebraic splines.

UO is in a unique position to offer training by leading experts both in algebraic geometry and in spline theory. The idea is to locate, with the help of the industrial partners Missler and SIM-A, appropriate geometric modelling or visualization problems that can be attacked through the theory of algebraic splines, using algebraic-geometric methods. The student would spend a longer period at INRIA, learning to use their libraries SYNAPS and AXEL, with the purpose of applying these tools to algebraic spline curves and surfaces, and would visit Missler or SIM-A, as well as VU-MIF, for short stays.

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