Towards a New Generation of Robust CAD/CAM Systems

本项目的目的是提高人们对复杂物体几何形状的稳健表示、查询和操作的知识水平，使用不精确/离散的计算机算法，并发展必要的理论和算法，为未来的CAD/CAM系统奠定基础。这一理论将主要涉及数值分析中的区间算术和条件概念，以及微分拓扑中的稳定性概念，代数拓扑中的单纯复理论，以及微分几何的基本概念，所有这些都应用于计算几何表示和算法中。具体目标是：分析如何使用区间非均匀有理B样条曲线和曲面片来获得复杂对象建模系统中实体的灵活自由形式边界的健壮机器实现。开发允许稳定布尔运算的方法。研究由布尔运算产生的邻接关系的局部稳定性性质。研究如何将布尔运算产生的邻接关系的局部稳定性性质集成到全局拓扑广义边界表示元组结构中，以获得一致的几何结果，并保持与可变数值分辨率的兼容性。对系统的结果信息不确定的情况和答案是具体的情况进行可靠的分类。

The object of this project is to advance the state of knowledge in robustly representing, interrogating and manipulating the geometry of complex objects using imprecise/discrete computer arithmetic and developing the necessary theory and algorithms which can form the basis for CAD/CAM systems of the future. This theory will involve primarily concepts of interval arithmetic and conditioning from numerical analysis as well as concepts of stability from differential topology, of simplicial complex theory from algebraic topology, and basic concepts of differential geometry, all applied in computational geometry representations and algorithms. Specific objectives are: To analyze how interval non-uniform rational B-spline curves and surface patches can be used to obtain robust machine realizations of flexible free-form boundaries for solids in a modeling system for complex objects. To develop methods which allow stable Boolean operations. To investigate how local stability properties of adjacencies resulting from Boolean operations. To investigate how local stability properties of adjacencies resulting from Boolean operations can be integrated in a global topological generalized Boundary Representation cell- tuple structure, so as to obtain consistent geometric results which remain compatible with a variable numerical resolution. To obtain a reliable classification of cases where the resulting information of the system is uncertain and those cases where the answer is specific.