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 (l(n)) and (L-n) of inscribed and circumscribed polygons, respectively. Numerical computations show that their converg ence is linear with the same convergence rate. This result is easy to prove for the parabola. For arbitrary de Rham curves, we prove two nea rby results. First, the existence of a limit q is an element of]0, 1[ of the sequence of ratios (Ln+1 - L)/(L-n - L) implies the convergence to the same limit of the two sequences (l(n+1) - L)/(l(n) - L) and (L n+1 - l(n+1))/(L-n - l(n)). Second, the sequence (Ln+1 - L-n) is bound ed by a convergent geometric sequence. In practice, this allows us to accelerate the convergence of both sequences by standard extrapolation algorithms. (C) 1998 Academic Press.