The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin’s series
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^...
| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Universidad de La Frontera
2019-08-01
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| Series: | Cubo |
| Subjects: | |
| Online Access: | http://revistas.ufro.cl/ojs/index.php/cubo/article/view/2158/1889 |
| Summary: | 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 $q=\frac{b}{a}\in[0,1]$. |
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| ISSN: | 0716-7776 0719-0646 |