Curve shortening by short rulers

Abstract

The aim is to construct a straight line between the two endpoints of a rectifiable curve using only a ruler, which is too short to connect them directly. The short ruler allows it to construct a polygonal line along the curve. Consecutive midpoints of it can be connected by the short ruler. That produces a shorter polygonal line and if the shortening is iterated, the polygonal line converges to the straight line. This short ruler method can be transported to cylinders. There, the ruler establishes short geodesics instead of straight lines. The produced sequence of curves converges to a geodesic which is homotopic to the original curve, but doesn’t have to be the shortest connection between the endpoints.

Keywords

• curve shortening
• short ruler method
• iteration
• geodesic
• cylinder