Curves With Known Arc Length I would appreciate if you could list as many (planar) curves with known closed-form analytical expressions for the arc length as possible. Please include formulas for both  the curve and the arc length.  The implicit curves  are of particular interest to me.

I might as well start the list : 


*

*Circle $S^1$


*

*implicit equation: $\quad\left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1, \quad$ parametrization: $\ \begin{cases} x = r\cos t, \\ y = r\sin t, \end{cases} \  t \in [0, 2\pi)$. 

*arc length $ s(t) =  r\cdot t, \ t \in [0, 2\pi)$, and $s(x,y) = r \cdot \arctan\left(\frac{y}{x}\right), \ 0\le x,y\leq r $.


*Parabola with focal length $f$, perpendicular distance to the axis of symmetry  $p$.


*

*implicit equation: $\left( x - h\right)^2 = 4 p \, (y-k)$.

*arc length from the vertex of parabola $s = \frac{hq}{f} + f \ln \left( \frac{h+q}{f}\right)$, $h = p/2$, $q = \sqrt{f^2 + h^2}$. 


*$ y =  x^2 - \frac{1}{8}\ln x $.


*

*arc length  from the point $(1,1)$: $\ s(x) =  x^2 + \frac{1}{8}\ln x - 1$.




PS: Please do not hesitate to post curves in higher dimensions.
 A: Catenary,
$f=\cosh(x)$,
since
$f'(x)
=\sinh(x)
$
so
$\sqrt{f'^2(x)+1}
=\sqrt{\sinh^2(x)+1}
=\cosh(x)
=f(x)
$.
A: *


*
Calculate the arc length of the graph of the function $y = x^{3/2}$ between the points $(0,0)$ and $(1,1)$.



*
Calculate the arc length of the graph of the function $y = (1/4) x^2-(1/2) \ln x$, between the points $(1,1/4)$ and $\bigl(e,(e^2-2)/4\bigr)$.



*
Calculate the arc length of the cycloid given by the parametric equations
\[
x(t) = t- \sin(t), \quad y(t) = 1-\cos(t) \quad \text{where} \quad 0 \leq t \leq 2 \pi.
\]



*
Calculate the arc length of the astroid curve given by the parametric equations
\[
x(t) = (\cos t)^3, \quad y(t) = (\sin t)^3 \quad \text{where} \quad 0 \leq  t \leq 2 \pi.
\]



*
Calculate the arc length of the spiral given by the parametric equations
\[
x(t) = (\exp t)(\cos t), \quad y(t) = (\exp t)(\sin t) \quad \text{where} \quad -\pi \leq t \leq \pi.
\]


A: The tractrix has parametric equations:
$$
x(t)=a(t-\tanh (t)) \qquad y(t)=a \mbox{ sech }( t)
$$
and the arc length is: $s(t)=a \ln (\cosh (t))$.
A curve with  implicit equation and simple arc lenght is the nephroid:
$$
108a^4x^2=(x^2+y^2-4a^2)^3
$$
that has  lenght: $L=24a$
And another one is the deltoid.
