Curves With Known Arc Length

Question

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.

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I might as well start the list :

1. 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 $.

2. 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 $s = \frac{hq}{f} + f \ln \left( \frac{h+q}{f}\right)$, $h = p/2$, $q = \sqrt{f^2 + h^2}$.

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

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

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PS: Please do not hesitate to post curves in higher dimensions.

Answer 1

Catenary, $f=\cosh(x)$, since $f'(x) =\sinh(x) $ so $\sqrt{f'^2(x)+1} =\sqrt{\sinh^2(x)+1} =\cosh(x) =f(x) $.

Answer 2

1. Calculate the arc length of the graph of the function $y = x^{3/2}$ between the points $(0,0)$ and $(1,1)$.
2. 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 $\bigl(e,(e^2-2)/4\bigr)$.
3. Calculate the arc length of the cycloid given by the parametric equations \[ x(t) = t- \sin(t), \quad y(t) = 1-\cos(t) \quad \text{where} \quad 0 \leq t \leq 2 \pi. \]
4. Calculate the arc length of the astroid curve given by the parametric equations \[ x(t) = (\cos t)^3, \quad y(t) = (\sin t)^3 \quad \text{where} \quad 0 \leq t \leq 2 \pi. \]
5. Calculate the arc length of the spiral given by the parametric equations \[ x(t) = (\exp t)(\cos t), \quad y(t) = (\exp t)(\sin t) \quad \text{where} \quad -\pi \leq t \leq \pi. \]

Answer 3

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.