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 $a\ge b\ge 0$ a sequence $Q_n(a,b)$ is constructed such that the relative error of the approximation $ P(a,b)\approx Q_n(a,b)$ satisfies the following inequalities \begin{align*} 0\le -\frac{P(a,b)-Q_n(a,b)}{P(a,b)}&\le \frac{(1-q^2)^{n+1}}{(2n+1)^2} \\ & \le \frac{1}{(2n+1)^2}\,e^{-q^2(n+1)}, \end{align*} true for $n\in\N$ and $q=\frac{b}{a}\in[0,1]$.

Keywords

• approximation
• elementary
• ellipse
• estimate
• maclaurin series
• mathematical validity
• perimeter
• simple