Curves, Circles, and Spheres

The standard radius of curvature at a point q(s) on a smooth curve can be defined as the limiting radius of circles through three points that all coalesce to q(s). In the study of ideal knot shapes it has recently proven useful to consider a global radius of curvature of the curve at q(s) defined as the smallest possible radius amongst all circles passing through this point and any two other points on the curve, coalescent or not. In particular, the minimum value of the global radius of curvature gives a convenient measure of curve thickness. Given the utility of the construction inherent to global curvature, it is also natural to consider variants of global radii of curvature defined in related ways. For example multi-point radius functions can be introduced as the radius of a sphere through four points on the curve, circles that are tangent at one point of the curve and intersect at another, etc. Then single argument, global radius of curvature functions can be constructed by minimizing over all but one argument. In this article we describe the interrelations between all possible global radius of curvature functions of this type, and show that there are two of particular interest. Properties of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices, including certain critical helices that arise in the optimal shapes of compact filaments, in α-helical proteins, and in B-form DNA.