Ellipse
Arc length

More generally, the arc length of a portion of the circumference, as a function of the angle subtended (or 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 y = b 1 − x 2 a 2. {\displaystyle y=b\ {\sqrt {1-{\frac {x^{2}}{a^{2}}}\ }}~.}

Then the arc length s {\displaystyle s} from x 1 {\displaystyle \ x_{1}\ } to x 2 {\displaystyle \ x_{2}\ } is: s = − b ∫ arccos x 1 a arccos x 2 a 1 + (a 2 b 2 − 1) sin 2 z d z. {\displaystyle 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~.}

This is equivalent to s = b z = arccos x 2 a arccos x 1 a {\displaystyle s=b\ \left_{z\ =\ \arccos {\frac {x_{2}}{a}}}^{\arccos {\frac {x_{1}}{a}}}}

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

Some lower and upper bounds on the circumference of the canonical ellipse x 2 / a 2 + y 2 / b 2 = 1 {\displaystyle \ x^{2}/a^{2}+y^{2}/b^{2}=1\ } with a ≥ b {\displaystyle \ a\geq b\ } are 2 π b ≤ C ≤ 2 π a , π (a + b) ≤ C ≤ 4 (a + b) , 4 a 2 + b 2 ≤ C ≤ 2 π a 2 + b 2. {\displaystyle {\begin{aligned}2\pi b&\leq C\leq 2\pi a\,\\\pi (a+b)&\leq C\leq 4(a+b)\,\\4{\sqrt {a^{2}+b^{2}\ }}&\leq C\leq {\sqrt {2\ }}\pi {\sqrt {a^{2}+b^{2}\ }}~.\end{aligned}}}

Here the upper bound 2 π a {\displaystyle \ 2\pi a\ } is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound 4 a 2 + b 2 {\displaystyle 4{\sqrt {a^{2}+b^{2}}}} is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and the minor axes.
detectable_format:multiple_sections[4]