A simple closed curve on the plane can be parameterized by a homeomorphism in infinitely many ways. It is natural to look for “nice” parameterizations: smooth ones, for example. I do not want to require smooth here, so let us try to find that is nonexpanding, that is for all . Note that Euclidean distance is used here, not arclength.
What are some necessary conditions for the existence of a nonexpanding parameterization?
- The curve must have length at most , since nonexpanding maps do not increase length. But this is not sufficient: an equilateral triangle of sidelength has no nonexpanding parameterization, despite its length being .
- The curve must have diameter at most 2 (which the triangle in item 1 fails). Indeed, nonexpanding maps do not increase the diameter either. However, is not sufficient either: an equilateral triangle of sidelength has no nonexpanding parameterization, despite its diameter being 2 (and length ).
- The curve must be contained in some closed disk of radius 1. This is not as obvious as the previous two items. We need Kirszbraun’s theorem here: any nonexpanding map extends to a nonexpanding map , and therefore is contained in the closed disk of radius 1 centered at . (This property fails for the triangle in item 2.)
The combination of 1 and 3 (with 2 being superseded by 3) still is not sufficient. A counterexample is given by any polygon that has length but is small enough to fit in a unit disk, for example:
Indeed, since the length is exactly , a nonexpanding parametrization must have constant speed 1. But mapping a circular arc onto a line segment with speed 1 increases pairwise Euclidean distances, since we are straightening out the arc.
Since I do not see a way to refine the necessary conditions further, let us turn to the sufficient ones.
- It is sufficient for to be a convex curve contained in the unit disk. Indeed, the nearest-point projection onto a convex set is a nonexpanding map, and projecting the unit circle onto the curve in this way gives the desired parameterization.
- It is sufficient for the curve to have length at most 4. Indeed, in this case there exists a parameterization with . For any the length of the shorter subarc between and is at most . Therefore, the length of is at most , which implies .
Clearly, neither of two sufficient conditions is necessary for the existence of a nonexpanding parameterization. But one can consider “length is necessary, length is sufficient” a reasonable resolution of the problem: up to some constant, the length of a curve can decide the existence one way or the other.
Stay tuned for a post on noncontracting parameterizations…