SGER: Topological Issues of Intersection Curves

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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.