SGER: Topological Issues of Intersection Curves

摘要0231511尼古拉斯M.我们提出研究平面和三维空间中覆盖相交曲线的有限盒集合的并集的拓扑。由于曲面相交而产生的曲线的表示几乎从不精确，因此采用了某些近似方案。其结果通常是一个分段线性近似cl，它位于实际曲线c的某个邻域N内。这一领域的研究主要集中在c的近似cl以及c和cl如何不相似。我们建议在盒的集合和交曲线上建立足够的条件，使得集合的并集是一个同伦等价于给定曲线的拓扑流形。在曲线没有自交点的情况下，这种集合可以作为曲线的另一种表示方法，因此这种新的表示方法可以用于各种工程应用中。我们认为这是一项适合SGER资助的创新探索性研究。

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.