Editing Ellipse (section)

===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>