On Quantitative Assessment of Chirality: Right- and Left-Handed Geometric Objects

Abstract

Two methods for quantitatively assessing the chirality of a set are considered. As a measure of the noncoincidence between two sets, one method uses the area of the symmetric difference between them, and the other, the Hausdorff distance between them. It is shown that these methods, generally speaking, do not provide a correct quantitative estimate for a fairly wide class of sets, such as bounded Borel sets. Using examples of flat triangles and convex quadrangles, we consider the problem of dividing geometric objects into right- and left-handed ones. For triangles, level lines of two versions of the chirality measure are calculated on the plane of angular parameters. For a spatial helix, the values of two versions of the chirality index are found by calculating the mixed product of vectors and the Hausdorff distance between two sets, respectively.