Mathematical Methods in Geometric Modelling

应用研究的许多领域都需要对曲线和曲面进行操作，然后进行后续分析。这些领域包括几何建模、计算机辅助设计（CAD）、计算机图形学和计算机视觉：CAD: CAD的一个典型应用涉及逆向工程，其中一个物理对象的模型被“输入”到计算机中，用于可能包括重新设计或更详细的数值分析的应用，例如，有限元方法。计算机视觉：自动导航车辆需要准确地解释导航图像，因此必须做出正确的决策。曲线与曲面交点的计算在这些应用中经常出现，这种运算简化为单变量多项式方程的求解。虽然这是应用数学中的一个经典问题，但它是一个重要的计算任务，因为浮点错误可能导致不满意的答案，而主要问题可能来自不准确的数据和不健壮的计算。因此，需要对实际数据执行令人满意的计算鲁棒算法是很重要的。建议的暑期学校将提供一系列讲座和计算机实验课，关于计算高次多项式的多重根的贬斥流形方法。如上所述，这种计算在实践中经常发生，因此建议的课程将引入一种新的数学技术，解决一些在几何建模和CAD系统中引起问题的问题。该方法是一种全新的方法，与现有方法有很大的不同，计算结果明显优于现有方法。

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.