The Length of the de Rham Curve

The length L of the de Rham curve is the common limit of two monotonic sequences of lengths ln‘ and Ln‘ of inscribed and circumscribed polygons, respectively. Numerical computations show that their convergence is linear with the same convergence rate. This result is easy to prove for the parabola. For arbitrary de Rham curves, we prove two nearby results. First, the existence of a limit q 2 “0; 1’ of the sequence of ratios LnC1 − L‘=Ln − L‘ implies the convergence to the same limit of the two sequences lnC1 − L‘=ln − L‘ and LnC1 − lnC1‘=Ln − ln‘. Second, the sequence LnC1 − Ln‘ is bounded by a convergent geometric sequence. In practice, this allows us to accelerate the convergence of both sequences by standard extrapolation algorithms.