TITLE: Curves With Known Arc Length
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.