SBIR Phase I: Applications of Morse Theory in Reverse Engineering

这个小企业创新研究第一阶段的项目提出了调查应用组合莫尔斯理论在逆向工程。它依赖于一个单一的数学方法：在多边形模型上定义一个连续函数，并根据该函数的梯度流分解表面。这个理论的变体将被用来解决几个重要的问题，包括自动转换成一组光滑的NURBS曲面，局部平滑多边形模型，并确定特征线的三角剖分。与早期的转换方法相比，这种方法的主要优点是能够将算法基于许多不同的表面分析标准，并通过混合这些不同的标准来实现最佳结果。莫尔斯理论是计算自然适应和遵循曲面形状的面片布局的关键，这是一个很难实现的属性，但对于自动构建扫描或三角化CAD模型的高质量NURBS曲面是必要的。 该项目的应用可以分为两种用途：o复制：参考物理部件创建数字模型。o重塑：设计的数字模型与物理参考保持一致。拟议的创新可能会对目前在设计和制造过程中没有使用数字流程的客户产生重大影响;这涵盖了美国制造的90%的产品。在全球制造业中，只有1%的产品使用CAD/CAM系统。有许多应用可以从成功实施该建议中受益，例如对实际零件的CFD分析，遗留零件的数字库存，历史保存以及定制消费品，包括医疗设备，鞋子，衣服和可穿戴计算机。

This Small Business Innovation Research Phase I project proposes to investigate applications of Combinatorial Morse Theory in Reverse Engineering. It relies on a single mathematical approach: the definition of a continuous function on a polygonal model and the decomposition of the surface based on the gradient flow of that function. Variants of this theory will be used to solve several important problems, including the automatic conversion of triangulations into a set of smooth NURBS surfaces, locally smoothing polygonal models, and identifying feature lines. The major advantage of this over earlier approaches to the conversion problem is the capacity to base the algorithm on a number of different criteria for surface analysis and to achieve the best result by intermingling these different criteria. Morse theory is the key to computing patch layouts that naturally adapt to and follow the shape of the surface, a property that is difficult to achieve but necessary to automatically construct high-quality NURBS surfaces of scanned or triangulated CAD models. Applications for the proposed project can be classified into two usages:o Duplication: referencing a physical part creates a digital model.o Remodeling: a designed digital model is kept to be consistent with a physical reference.The proposed innovation may have significant impact on customers who are not using a digital process today in their design and manufacturing process; this covers 90% of all products being made in the United States. Worldwide manufacturing, only 1% of all products made uses CAD/CAM systems. There are many applications that can benefit from a successful implementation of this proposal, such as CFD analysis on actual parts, digital inventory for legacy parts, historical preservation, and custom-made consumer products including medical devices, shoes, clothes, and wearable computers.