Editing Ellipse (section)

== Metric properties ==
All metric properties given below refer to an ellipse with equation
{{NumBlk2||<math display="block">\frac{x^2}{a^2}+\frac{y^2}{b^2}= 1 </math>|1}}
except for the section on the area enclosed by a tilted ellipse, where the generalized form of Eq.({{EquationNote|1}}) will be given.

=== Area ===
The [[area]] <math>A_\text{ellipse}</math> enclosed by an ellipse is:
{{NumBlk2||<math display="block">A_\text{ellipse} = \pi ab</math>|2}}

where <math>a</math> and <math>b</math> are the lengths of the semi-major and semi-minor axes, respectively.  The area formula <math>\pi a b</math> is intuitive: start with a circle of radius <math>b</math> (so its area is <math>\pi b^2</math>) and stretch it by a factor <math>a/b</math> to make an ellipse. This scales the area by the same factor: <math>\pi b^2(a/b) = \pi a b.</math><ref>{{Cite book|last=Archimedes.|url=https://www.worldcat.org/oclc/48876646|title=The works of Archimedes|date=1897|publisher=Dover Publications|others=Heath, Thomas Little, Sir, 1861-1940. | isbn=0-486-42084-1 | location=Mineola, N.Y.|pages=115|oclc=48876646}}</ref> However, using the same approach for the circumference would be fallacious – compare the [[integral]]s <math display="inline">\int f(x)\, dx</math> and <math display="inline"> \int \sqrt{1+f'^2(x)}\, dx</math>. It is also easy to rigorously prove the area formula using integration as follows.  Equation ({{EquationNote|1}}) can be rewritten as <math display="inline">y(x) = b \sqrt{1 - x^2 / a^2}.</math>  For <math>x\in[-a,a],</math> this curve is the top half of the ellipse.  So twice the integral of <math>y(x)</math> over the interval <math>[-a,a]</math> will be the area of the ellipse:
<math display="block">\begin{align}
  A_\text{ellipse} &= \int_{-a}^a 2b\sqrt{1 - \frac{x^2}{a^2}}\,dx\\
                   &= \frac ba \int_{-a}^a 2\sqrt{a^2 - x^2}\,dx.
\end{align}</math>

The second integral is the area of a circle of radius <math>a,</math> that is, <math>\pi a^2.</math> So
<math display="block">A_\text{ellipse} = \frac{b}{a}\pi a^2 = \pi ab.</math>

An ellipse defined implicitly by <math>Ax^2+ Bxy + Cy^2 = 1 </math> has area <math>2\pi / \sqrt{4AC - B^2}.</math>

The area can also be expressed in terms of eccentricity and the length of the semi-major axis as <math>a^2\pi\sqrt{1-e^2}</math> (obtained by solving for [[flattening]], then computing the semi-minor axis).

[[Image:tiltedEllipse2.jpg|thumb|The area enclosed by a tilted ellipse is <math>\pi\; y_\text{int}\, x_\text{max}</math>.]]

So far we have dealt with ''erect'' ellipses, whose major and minor axes are parallel to the <math>x</math> and <math>y</math> axes. However, some applications require ''tilted'' ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its ''emittance''. In this case a simple formula still applies, namely
{{NumBlk2||<math display="block">A_\text{ellipse} = \pi\; y_\text{int}\, x_\text{max} = \pi\; x_\text{int}\, y_\text{max}</math>|3}}

where <math>y_{\text{int}}</math>, <math>x_{\text{int}}</math> are intercepts and <math>x_{\text{max}}</math>, <math>y_{\text{max}}</math> are maximum values. It follows directly from [[Ellipse#Theorem of Apollonios on conjugate diameters|Apollonios's theorem]].

===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}}

===Arc length===
{{further|Meridian arc#Calculation}}

More generally, the [[arc length]] of a portion of the circumference, as a function of the angle subtended (or {{nobr|{{mvar|x}} coordinates}} of any two points on the upper half of the ellipse), is given by an incomplete [[elliptic integral]]. The upper half of an ellipse is parameterized by
<math display="block"> y = b\ \sqrt{ 1-\frac{x^{2}}{a^{2}}\ } ~.</math>

Then the arc length <math>s</math> from <math>\ x_{1}\ </math> to <math>\ x_{2}\ </math> is:
<math display="block">s = -b\int_{\arccos \frac{x_1}{a}}^{\arccos \frac{x_2}{a}} \sqrt{\ 1 + \left( \tfrac{a^2}{b^2} - 1 \right)\ \sin^2 z ~} \; dz ~.</math>

This is equivalent to
<math display="block"> s = b\ \left[ \; E\left(z \;\Biggl|\; 1 - \frac{a^2}{b^2} \right) \; \right]^{\arccos \frac{x_1}{a}}_{z\ =\ \arccos \frac{x_2}{a}} </math>

where <math>E(z \mid m)</math> is the incomplete elliptic integral of the second kind with parameter <math>m=k^{2}.</math>

Some lower and upper bounds on the circumference of the canonical ellipse <math>\ x^2/a^2 + y^2/b^2 = 1\ </math> with <math>\ a \geq b\ </math> are<ref>{{cite journal |last1=Jameson |first1=G.J.O. |year=2014 |title=Inequalities for the perimeter of an ellipse |journal= Mathematical Gazette |volume=98 |issue=542 |pages=227–234 |doi=10.1017/S002555720000125X |s2cid=125063457}}</ref>
<math display="block">\begin{align}
             2\pi b &\le C \le 2\pi a\ , \\
          \pi (a+b) &\le C \le 4(a+b)\ , \\
  4\sqrt{a^2+b^2\ } &\le C \le \sqrt{2\ } \pi \sqrt{a^2 + b^2\ } ~.
\end{align}</math>

Here the upper bound <math>\ 2\pi a\ </math> is the circumference of a [[circumscribed circle|circumscribed]] [[concentric circle]] passing through the endpoints of the ellipse's major axis, and the lower bound <math>4\sqrt{a^2+b^2}</math> is the perimeter of an [[inscribed figure|inscribed]] [[rhombus]] with [[vertex (geometry)|vertices]] at the endpoints of the major and the minor axes.

Given an ellipse whose axes are drawn, we can construct the endpoints of a particular elliptic arc whose length is one eighth of the ellipse's circumference using only [[Straightedge and compass construction|straightedge and compass]] in a finite number of steps; for some specific shapes of ellipses, such as when the axes have a length ratio of {{tmath|\sqrt2 : 1}}, it is additionally possible to construct the endpoints of a particular arc whose length is one twelfth of the circumference.<ref>{{Cite book |last1=Prasolov |first1=V. |last2=Solovyev|first2=Y.|title=Elliptic Functions and Elliptic Integrals|publisher=American Mathematical Society |year=1997 |isbn=0-8218-0587-8|pages=58–60}}</ref> (The vertices and co-vertices are already endpoints of arcs whose length is one half or one quarter of the ellipse's circumference.) However, the general theory of straightedge-and-compass elliptic division appears to be unknown, unlike in [[Constructible polygon|the case of the circle]] and [[Lemniscate elliptic functions|the lemniscate]]. The division in special cases has been investigated by [[Adrien-Marie Legendre|Legendre]] in his classical treatise.<ref>Legendre's ''Traité des fonctions elliptiques et des intégrales eulériennes''</ref>

=== Curvature ===
The [[curvature]] is given by:

<math display="block">\kappa = \frac{1}{a^2 b^2}\left(\frac{x^2}{a^4}+\frac{y^2}{b^4}\right)^{-\frac{3}{2}}\ ,</math>

and the [[Radius of curvature#Ellipses|radius of curvature]], ρ = 1/κ, at point <math>(x,y)</math>:
<math display="block">\rho = a^2 b^2 \left(\frac{x^{2}}{a^4} + \frac{y^{2}}{b^4}\right)^\frac{3}{2} = \frac{1}{a^4 b^4} \sqrt{\left(a^4 y^{2} + b^4 x^{2}\right)^3} \ .</math>The radius of curvature of an ellipse, as a function of angle ''{{mvar|θ}}'' from the center, is:
<math display="block">R(\theta)=\frac{a^2}{b}\biggl(\frac{1-e^2(2-e^2)(\cos\theta)^2)}{1-e^2(\cos\theta)^2}\biggr)^{3/2}\,,</math>where e is the eccentricity.

Radius of curvature at the two ''vertices'' <math>(\pm a,0)</math> and the centers of curvature:
<math display="block">\rho_0 = \frac{b^2}{a}=p\ , \qquad \left(\pm\frac{c^2}{a}\,\bigg|\,0\right)\ .</math>

Radius of curvature at the two ''co-vertices'' <math>(0,\pm b)</math> and the centers of curvature:
<math display="block">\rho_1 = \frac{a^2}{b}\ , \qquad \left(0\,\bigg|\,\pm\frac{c^2}{b}\right)\ .</math>The locus of all the centers of curvature is called an [[Evolute#Evolute of an ellipse|evolute]]. In the case of an ellipse, the evolute is an [[astroid]].