Editing Ellipse (section)

===Circumference===
{{Main|Perimeter of an ellipse}}
{{further|Meridian arc#Quarter meridian}}
[[File:Ellipses same circumference.png|thumb|Ellipses with same circumference]]

The circumference <math>C</math> of an ellipse is:
<math display="block">C \,=\, 4a\int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta \,=\, 4 a \,E(e)</math>

where again <math>a</math> is the length of the semi-major axis, <math display="inline">e=\sqrt{1 - b^2/a^2}</math> is the eccentricity, and the function <math>E</math> is the [[complete elliptic integral of the second kind]],
<math display="block">E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta</math>
which is in general not an [[elementary function]].

The circumference of the ellipse may be evaluated in terms of <math>E(e)</math> using [[arithmetic-geometric mean|Gauss's arithmetic-geometric mean]];<ref>{{dlmf|first=B. C.|last=Carlson|id=19.8.E6|title=Elliptic Integrals}}</ref> this is a quadratically converging iterative method (see [[Elliptic integral#Computation|here]] for details).

The exact [[infinite series]] is:
<math display="block">\begin{align}
  \frac C{2\pi a} &= 1 - \left(\frac{1}{2}\right)^2e^2 - \left(\frac{1\cdot 3}{2\cdot 4}\right)^2\frac{e^4}{3} - \left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)^2\frac{e^6}{5} - \cdots \\
    &= 1 - \sum_{n=1}^\infty \left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{e^{2n}}{2n-1} \\
    &= -\sum_{n=0}^\infty \left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{e^{2n}}{2n-1},
\end{align}
</math>
where <math>n!!</math> is the [[double factorial]] (extended to negative odd integers in the usual way, giving <math>(-1)!! = 1</math> and <math>(-3)!! = -1</math>).

This series converges, but by expanding in terms of <math>h = (a-b)^2 / (a+b)^2,</math> [[James Ivory (mathematician)|James Ivory]],<ref>{{cite journal
|last = Ivory
|first = J.
|title = A new series for the rectification of the ellipsis
|author-link = James Ivory (mathematician)
|journal = Transactions of the Royal Society of Edinburgh
|year = 1798
|volume = 4
|issue = 2
|pages = 177{{ndash}}190
|url =https://books.google.com/books?id=FaUaqZZYYPAC&pg=PA177
|doi=10.1017/s0080456800030817
|s2cid = 251572677
}}</ref> [[Friedrich Wilhelm Bessel|Bessel]]<ref>{{cite journal
|ref = {{harvid|Bessel|1825}}
|last = Bessel
|first = F. W.
|title = The calculation of longitude and latitude from geodesic measurements (1825)
|author-link = Friedrich Bessel
|journal = [[Astron. Nachr.]]
|year = 2010
|volume = 331
|number = 8
|pages = 852{{ndash}}861
|arxiv = 0908.1824
|doi = 10.1002/asna.201011352
|bibcode = 2010AN....331..852K
|s2cid = 118760590
}} English translation of {{cite journal
 |first1= F. W. | last1=Bessel
 | doi=10.1002/asna.18260041601
 | year=1825
 | bibcode=1825AN......4..241B
 |title=Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermesssungen
 | journal=[[Astron. Nachr.]] |volume=4 |issue=16 |pages =241{{ndash}}254
 |arxiv=0908.1823 | s2cid=118630614 | language=de
}}</ref> and [[Ernst Kummer|Kummer]]<ref name=Linderholm95>{{cite journal
 |title=An Overlooked Series for the Elliptic Perimeter
 |first1=Carl E. |last1=Linderholm   |first2=Arthur C. |last2=Segal
 |journal=Mathematics Magazine |volume=68 |issue=3 |pages=216–220
 |date=June 1995
 |doi=10.1080/0025570X.1995.11996318
}} which cites to {{cite journal
 |last=Kummer |first=Ernst Eduard |author-link=Ernst Kummer
 |title=Uber die Hypergeometrische Reihe |language=de
 |trans-title=About the hypergeometric series
 |journal=[[Journal für die Reine und Angewandte Mathematik]]
 |volume=15 |issue=1, 2 |year=1836 |pages=39–83, 127–172
 |doi=10.1515/crll.1836.15.39
 |url=https://archive.org/details/sim_journal-fuer-die-reine-und-angewandte-mathematik_1836_15
}}</ref> derived a series that converges much more rapidly.  It is most concisely written in terms of the [[Binomial coefficient#Binomial coefficient with n = 1/2|binomial coefficient with <math>n = 1/2</math>]]:
<math display="block">\begin{align}
  \frac{C}{\pi(a+b)}
    &= \sum_{n=0}^\infty {\frac 12 \choose n}^2 h^n \\
    &= \sum_{n=0}^\infty \left(\frac{(2n-3)!!}{(2n)!!}\right)^2 h^n \\
    &= \sum_{n=0}^\infty \left(\frac{(2n-3)!!}{2^n n!}\right)^2 h^n \\
    &= \sum_{n=0}^\infty \left(\frac{1}{(2n-1)4^n}\binom{2n}{n}\right)^2 h^n \\
    &= 1 + \frac{h}{4} + \frac{h^2}{64} + \frac{h^3}{256} + \frac{25\,h^4}{16384} + \frac{49\,h^5}{65536} + \frac{441\,h^6}{2^{20}} + \frac{1089\,h^7}{2^{22}} + \cdots.
\end{align}</math>
The coefficients are slightly smaller (by a factor of <math>2n-1</math>), but also <math>e^4/16 \le h \le e^4</math> is numerically much smaller than <math>e</math> except at <math>h = e = 0</math> and <math>h = e = 1</math>.  For eccentricities less than 0.5 {{nobr|(<math>h < 0.005</math>),}} the error is at the limits of [[double-precision floating-point]] after the <math>h^4</math> term.<ref name=Cook23>{{cite web
 |title=Comparing approximations for ellipse perimeter
 |date=28 May 2023
 |first=John D. |last=Cook
 |website=John D. Cook Consulting blog
 |url=https://www.johndcook.com/blog/2023/05/28/approximate-ellipse-perimeter/
 |access-date=2024-09-16
}}</ref>

[[Srinivasa Ramanujan]] gave two close [[approximations]] for the circumference in §16 of "Modular Equations and Approximations to <math>\pi</math>";<ref>{{cite journal
|last=Ramanujan |first=Srinivasa |author-link=Srinivasa Ramanujan
|title=Modular Equations and Approximations to ''π''
|journal = Quart. J. Pure App. Math.
|volume = 45
|pages = 350{{ndash}}372
|year = 1914
|url = http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf#page=24
|isbn = 978-0-8218-2076-6
}}</ref> they are
<math display="block">\frac C\pi \approx 3(a + b) - \sqrt{(3a + b)(a + 3b)}  = 3(a + b) - \sqrt{3(a+b)^2 + 4ab}</math>
and
<math display="block">\frac C{\pi(a+b)} \approx 1+\frac{3h}{10+\sqrt{4-3h}},</math>
where <math>h</math> takes on the same meaning as above. The errors in these approximations, which were obtained empirically, are of order <math>h^3</math> and <math>h^5,</math> respectively.<ref name=Villarino>{{cite arXiv |eprint=math.CA/0506384 |title=Ramanujan's Perimeter of an Ellipse |first=Mark B. |last=Villarino |date=20 June 2005 |quote=We present a detailed analysis of Ramanujan’s most accurate approximation to the perimeter of an ellipse.}}  In particular, the second equation underestimates the circumference by <math>\pi(a+b)h^5\theta(h),</math> where <math>22.888\cdot 10^{-6} < 3\cdot 2^{-17} < \theta(h) \le 4\left(1 - \frac{7\pi}{22}\right) < 1.60935\cdot10^{-3}</math> is an increasing function of <math>0 \le h \le 1.</math></ref><ref>{{cite web
 |title=Error in Ramanujan's approximation for ellipse perimeter
 |date=22 September 2024
 |first=John D. |last=Cook
 |website=John D. Cook Consulting blog
 |url=https://www.johndcook.com/blog/2024/09/22/ellipse-perimeter-approx/
 |access-date=2024-12-01
 |quote=the relative error when {{math|''b'' {{=}} 1}} and {{mvar|a}} varies ... is bound by {{math|4/''π'' − 14/11 {{=}} 0.00051227…}}.
}}</ref>  This is because the second formula's infinite series expansion matches Ivory's formula up to the <math>h^4</math> term.{{r|Villarino|p=3}}