SGER: Topological Issues of Intersection Curves

We propose to study the topology of the union of a finite collection of boxes that covers an intersection curve in the plane and in 3D space. The representation of curves that are the result of surface intersections is nearly never exact, and thus certain approximations schemes are employed. The result is often a piecewise linear approximation cl, which lies within a certain neighborhood N, of the actual curve c. Much of the research in this area has been focused on the approximation cl of c and how c and cl are/are not similar. We propose to develop sufficient conditions on the collection of the boxes and the intersection curve, so that the resulting union of the collection is a topological manifold homotopically equivalent to the given curve. In the case where the curve has no self-intersections, this collection can serve as another means of representing the curve, and thus this new representation can be used in a variety of engineering applications.