SGER: Topological Issues of Intersection Curves

SGER：交集曲线的拓扑问题

• 批准号: 0231511
• 负责人: Nicholas Patrikalakis
• 金额: $ 5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-10-01 至 2005-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0231511&HistoricalAwards=false
• 关键词: SGER; Topological; Issues; Intersection; Curves

ABSTRACT0231511Nicholas M. PattrikalakisMass Inst. of TechWe propose to study the topology of the union of a finite collection of boxes that covers anintersection curve in the plane and in 3D space. The representation of curves that are theresult of surface intersections is nearly never exact, and thus certain approximations schemes areemployed. The result is often a piecewise linear approximation cl, which lies within a certainneighborhood N, of the actual curve c. Much of the research in this area has been focused onthe approximation cl of c and how c and cl are/are not similar. We propose to develop suficientconditions on the collection of the boxes and the intersection curve, so that the resulting unionof the collection is a topological manifold homotopically equivalent to the given curve. In thecase where the curve has no self-intersections, this collection can serve as another means ofrepresenting the curve, and thus this new representation can be used in a variety of engineeringapplications. We believe this to be an innovative exploratory study appropriate for SGERfunding.

尼古拉斯·M·帕特里卡拉基斯马萨诸塞州。我们建议研究覆盖平面和3D空间中相交曲线的有限个盒集合的并的拓扑。作为曲面交点结果的曲线的表示几乎从来都不是精确的，因此采用了某些近似方案。其结果通常是位于实际曲线C的某个邻域N内的分段线性逼近CL。这一领域的许多研究都集中在C的逼近CL以及C和CL如何相似/不相似上。我们给出了盒和相交曲线的集合的充分条件，使得得到的集合的单位是与给定曲线同伦等价的拓扑流形。在曲线没有自交点的情况下，这个集合可以作为另一种表示曲线的方法，因此这种新的表示可以在各种工程应用中使用。我们认为这是一项创新的探索性研究，适合为SGER提供资金。