TITLE: Explicit nontrivial examples of arc length parametrization
QUESTION [8 upvotes]: For pedagogical purposes (and for some numerical experiments) I was looking for some nontrivial explicit (i.e. closed-form) examples of arclength parametrized curves. I know that arclength parametrization always exists for a regular curve, but it seems that the general consensus is that trying to compute this explicitly is generally madness.
Of course any straight line and circular arc can be explicitly parametrized this way, but I was wondering if there are any other known nontrivial examples (especially ones with nonconstant curvature)?

REPLY [5 votes]: After some time I came across a few more examples to add to the list.

The logarithmic spiral,
$$
\alpha(t) = (e^t\cos t, e^t\sin t)
$$
which has $\|\alpha'(t)\|=\sqrt{2}\,e^t$ and so has arc length
$$
s = \int_0^t \|\alpha'(u)\|\,\mathrm{d}u =\int_0^t \sqrt{2}\,e^u \,\mathrm{d}u = \sqrt{2}\,(e^t-1)
$$
and so $t=\ln\left(\frac{s}{\sqrt{2}}+1\right)$ which let's us reparametrize the curve as
$$
\tilde\alpha(s)=\left( \left(\frac{s}{\sqrt{2}}+1\right)\cos\left(\ln\left(\frac{s}{\sqrt{2}}+1\right)\right), \left(\frac{s}{\sqrt{2}}+1\right)\sin\left(\ln\left(\frac{s}{\sqrt{2}}+1\right) \right)\right)
$$

The helix,
$$
\beta(t) = (a\cos t, a\sin t, bt)
$$
which has $\|\beta'(t)\| = \sqrt{a^2+b^2}$ and so has arc length
$$
s = \int_0^t \|\beta'(u)\|\,\mathrm{d}u\int_0^t \sqrt{a^2+b^2}\,\mathrm{d}u = t\sqrt{a^2+b^2}
$$
and so $t=\frac{s}{\sqrt{a^2+b^2}}$ which let's us reparametrize the curve as
$$
\tilde{\beta}(s) = \left( a\cos\left(\frac{s}{\sqrt{a^2+b^2}}\right), a\sin\left(\frac{s}{\sqrt{a^2+b^2}}\right), \frac{bs}{\sqrt{a^2+b^2}}\right)
$$

A helix-like curve inside in a flat torus,
$$
\gamma(t) = (a\cos At, a\sin At, b\cos Bt, b\sin Bt)
$$
which has $\|\gamma'(t)\|=\sqrt{a^2A^2+b^2B^2}$ and so has arc length
$$
s = \int_0^t \|\gamma'(u)\|\,\mathrm{d}u\int_0^t \sqrt{a^2A^2+b^2B^2}\,\mathrm{d}u = t\sqrt{a^2A^2+b^2B^2}
$$
and so $t=\frac{s}{C}$ where $C = \sqrt{a^2A^2+b^2B^2}$. This let's us reparametrize the curve as
$$
\tilde\gamma(s)=\left( a\cos\frac{As}{C}, a\sin\frac{As}{C},b\cos\frac{Bs}{C},b\sin\frac{Bs}{C}\right)
$$