The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin’s series

Abstract

For the perimeter P (a, b) of an ellipse with the semi-axes a ≥ b ≥ 0 a sequence Q n (a, b) is constructed such that the relative error of the approximation P (a, b) ≈ Q n (a, b) satisfies the following inequalities 0 ≤ − P (a, b) − Q n (a, b) P (a, b) ≤ (1 − q 2) n+1 (2n + 1) 2 ≤ 1 (2n + 1) 2 e −q 2 (n+1) , true for n ∈ N and q = b a ∈ [0, 1]. RESUMEN Para el perímetro P (a, b) de una elipse con semiejes a ≥ b ≥ 0, se construye una sucesión Q n (a, b) tal que el error relativo de la aproximación P (a, b) ≈ Q n (a, b) satisface las siguientes desigualdades 0 ≤ − P (a, b) − Q n (a, b) P (a, b) ≤ (1 − q 2) n+1 (2n + 1) 2 ≤ 1 (2n + 1) 2 e −q 2 (n+1) , válidas para n ∈ N y q = b a ∈ [0, 1].