ctext 割圜連比例術圖解 (清)董祐誠 wikipedia Larcombe-The_18th_century_Chinese_discovery_of_the_Catalan_numbers.pdf 明安圖和他的冪級數展開式 羅見今 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 $ {\displaystyle y=b\ {\sqrt {1-{\frac {x^{2}}{a^{2}}}\ }}~.} $ Then the arc length $ s $ from $ \ x_{1}\ $ to $ \ x_{2}\ $ is: $ {\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 $ {\displaystyle s=b\ \left[\;E\left(z\;{\Biggl |}\;1-{\frac {a^{2}}{b^{2}}}\right)\;\right]_{z\ =\ \arccos {\frac {x_{2}}{a}}}^{\arccos {\frac {x_{1}}{a}}}} $ where $ E(z\mid m) $ is the incomplete elliptic integral of the second kind with parameter $ m=k^{2}. $ Some lower and upper bounds on the circumference of the canonical ellipse $ \ x^{2}/a^{2}+y^{2}/b^{2}=1\ $ with $ \ a\geq b\ $ are[20] $ {\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\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{\sqrt {a^{2}+b^{2}}} $ is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and the minor axes.