Mathematical Methods in Geometric Modelling

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Many areas of applied research require the manipulation of curves and surfaces, which are then required for subsequent analysis. These areas include geometric modelling, computer-aided design (CAD), computer graphics and computer vision:CAD: A typical application of CAD involves reverse engineering, in which a model of a physical object is 'fed' into a computer for applications that may include redesign or more detailed numerical analysis by, for example, the finite element method.Computer vision: An autonomous guided vehicle requires the accurate interpretation of images for navigation, and it is therefore imperative that correct decisions are made.The calculation of the point(s) of intersection of curves and surfaces arises frequently in these applications, and this operation reduces to the solution of a univariate polynomial equation. Although this is a classical problem in applied mathematics, it is a non-trivial computational task because floating point errors can lead to unsatisfactory answers, and major problems can arise from inaccurate data and calculations that are not robust. The need for computationally robust algorithms that perform satisfactorily with real data is therefore important.The proposed Summer School will offer a series of lectures and computer laboratory classes on the method of pejorative manifolds for the computation of multiple roots of high degree polynomials. As noted above, this computation occurs frequently in practice, and thus the proposed course will introduce a new mathematical technique that addresses some of the issues that cause problems in geometric modelling and CAD systems. This method, which is very new, is radically different from existing methods, and computational results that are significantly better than those obtained by these methods have been achieved.