On the shapes of tangled curves
A new method is introduced for investigating the shapes of space curves. The method involves analyzing intersections between a curve and a moving manifold. The positions and orientations of the manifold are determined by the curve itself, and so the method is invariant under curve translations and rotations.
The method is investigated in detail in two instances, one in which the manifold is a planar triangle and the other in which the manifold is a disk. Each of these cases involves unique subtleties that are explored in depth. In all instances of the method, the intersections between manifold and curve are illustrated as a two dimensional point set. This is the power of the method, that it allows intricately convoluted space curves to be represented as a single digestible planar image. Whether one segment of curve wraps around or encircles another is conveyed by the presence of a line in the plane, no matter how deeply buried in a tangle these segments may be. Other aspects of the way in which segments fit together also follow from the method, such as whether the encirclement of one segment about another is tight or loose.
Although the method provides topological information (such as the linking number of two curves), we pursue it more for what it tells about curve shape and geometry. The planar image generated by the method rapidly communicates information about curve structure, and so the image can be used as a shape descriptor, that is, as an abbreviation of critical shape parameters. As shape descriptors, images from the method can be used to rapidly compare and categorize libraries of curve structures. As a proof of concept demonstration, the method is used in this way to navigate a representative sample of protein backbone curves. The method successfully distinguishes between protein molecules from different structural families.
0548: Mechanical engineering