An Investigation on Surface-to-Surface Intersection Problems

这项工作的目的是研究一种新的方法来发现和描述在曲面到曲面相交的背景下产生的未知代数曲线的所有特征。我们的方法包括使用代数几何技术对曲线的方程进行符号推导，并在矩形窗口内以Bernstein基表示曲线，这允许分段表示并为底层系数提供几何意义。提出了一种新的计算曲线特征点的方法，该方法有助于保证正确的拓扑结构。它们基于符号计算、用于大规模级数求和的自适应方法、以提高精度和最小化技术。还提出了基于非均匀自适应细分的曲线查询，以及基于Bernstein表示的性质的具有指定距离误差的隐式曲线的点和参数近似的推导。这项拟议工作的意义在于，它承诺提供强大的曲面到曲面相交能力，这在复杂文物的设计、分析和制造的更自动化模式中是必要的。我们技术的许多元素也是可并行化的，这是实时大规模应用的一个重要属性。

The objective of this work is to investigate a new method to discover and describe all features of an unknown algebraic curve resulting in the context of surface to surface intersections. Our method involves symbolic derivation of the equation of the curve, using algebraic geometry techniques, and its representation in the Bernstein basis within a rectangular window, which allows piecewise representation and provides geometrical significance to the underlying coefficients. Novel methods to compute the characteristic points of the curve, useful in guaranteeing correct topology, are proposed. These are based on symbolic computation, adaptive methods for large scale series summation, to enhance accuracy, and minimization techniques. Interrogation of the curve, based on non-uniform adaptive subdivision, and derivation of point and parametric approximations of the implicit curve, with prescribed distance errors, based on the properties of the Bernstein representation, are also proposed for study. The significance of the proposed work is that it promises to provide a robust surface to surface intersection capability, which is necessary in a more automated mode of design, analysis and manufacturing of complex artifacts. Many elements of our technique are also parallelizable which is an important attribute for real time large scale applications.