How is the arc length of an ellipse (measured from the vertex) defined by $x = a \cos (\theta)$, $y = b \sin(\theta)$ given by $s(\psi) = a Elliptic\left(\psi,\sqrt{1-\frac{b^2}{a^2}}\right) $. Please see my attempt below
\begin{align*}
s(\psi) &~=~\int_{0}^{\psi} \sqrt{\left(\frac{dx}{d\theta}\right)^2+ \left(\frac{dy}{d\theta}\right)^2} d\theta
\\
&~=~ \int_{0}^{\psi} \sqrt{ a^2 \sin^2(\theta)+b^2\cos^2(\theta)} d\theta
\\
&~=~ \int_{0}^{\psi} \sqrt{ a^2 (1-\cos^2(\theta))+b^2\cos^2(\theta)} d\theta
\\
&~=~ \int_{0}^{\psi} \sqrt{ a^2+(b^2-a^2)\cos^2(\theta)} d\theta
\\
&~=~ a \int_{0}^{\psi} \sqrt{ 1-\left(1-\frac{a^2}{b^2}\right)\cos^2(\theta)} d\theta,
\end{align*}
which is not equal to
\begin{align*}
s(\psi) &~=~ a Elliptic\left(\psi,\sqrt{1-\frac{b^2}{a^2}}\right)
\\
&~=~a \int_{0}^{\psi} \sqrt{ 1-\left(1-\frac{b^2}{a^2}\right)\sin^2(\theta)}.
\end{align*}
Arc length of a point on ellipse from the vertex
arc lengthcalculusconic sectionsintegrationtrigonometry
Best Answer
For clockwise sense:
\begin{align} (x,y) &= (a\sin \theta,b\cos \theta) \\ s(\theta) &= aE(\theta, \varepsilon) \\ \varepsilon &= \sqrt{1-\frac{b^2}{a^2}} \end{align}
For anti-clockwise sense:
\begin{align} (x,y) &= (a\cos \theta,b\sin \theta) \\ s(\theta) &= bE\left( \theta, \frac{ia\varepsilon}{b} \right) \\ &= aE\left( \frac{\pi}{2}-\theta, \varepsilon \right) \\ \frac{ia\varepsilon}{b} &= \sqrt{1-\frac{a^2}{b^2}} \end{align}
Parametrized with Jacobi elliptic functions:
\begin{align} (x,y) &= (a\operatorname{sn} (u,\varepsilon), b\operatorname{cn} (u,\varepsilon)) \\ s(u) &= aE( \operatorname{am} ( u,\varepsilon ), \varepsilon ) \\ \end{align}