From oscillating speed along curve to constant speed along same curve Let a parameterization of the curve
$$
\{(x,y)\in \mathbb{R}^2: x=y, \|(x,y)\|\leq 1\}
$$
be $\{\alpha(t)\}_{t\in [0,t^\alpha]}$ such that $\alpha(t) = (1,1)*(-cos(t^2))$. Determine from this parameterization another parameterization $\{\beta(t)\}_{t\in [0,t^\beta]}$ such that the corresponding speed along the curve $\{\{\|\beta'(t)\|\}_{t\in [0,t^\beta]}$ is constant $s$. That is, determine a final time $t^\beta$ and a function $f$ such that $\beta = \alpha \circ f$, $f(0) = 0$, and for every $t$ in $[0,t^\beta]$, $s=\|(\alpha\circ f)'(t)\|$. It is that
\begin{align}
 s&=\|(\alpha\circ f)'(t)\|\\
&= \|\alpha'(f(t))f'(t)\|\\
&=\|\alpha'(f(t))\||f'(t)|\\
&=\|(1,1)\sin(f(t)^2)2f(t)\||f'(t)|\\
\end{align}
Hence, this is a differential equaion in terms of $f$. When I try to numerically solve this it appears that I get into trouble due to the $\|(1,1)\sin(f(t)^2)2f(t)\|$ periodically assuming the value zero, which then forces |f'(t)| to be infinite. This occurs at the endpoints of the curve, where the velocity changes direction instaenously. Maybe I am missing a specification on the derivative of $\alpha\circ f$ that defines this change in velocity?
 A: The theory is, if $\alpha$ is a regular parametrization (continuously-differentiable with non-vanishing velocity) on some interval $[0, t^{\alpha}]$, and if we define the arclength function $s$ by
$$
s(t) = \int_{0}^{t} \|\alpha'\|
$$
and set $\ell = s(t^{\alpha})$, then $s:[0, t^{\alpha}] \to [0, \ell]$ has a continuously-differentiable inverse since $s' = \|\alpha'\|$ is strictly positive, and the path $\beta = \alpha \circ s^{-1}$ has unit speed.
As you note, there is a theoretical obstacle in the proposed example: The velocity vanishes at the endpoint $\sqrt{\pi/2}$, so $\alpha$ is not a regular path and the inverse mapping is not differentiable at the corresponding endpoint. If we pick a positive upper bound $t^{\alpha} < \sqrt{\pi/2}$, however, things work out as expected.

The remarks below may not apply to your situation, but are included for posterity. In general, practical obstacles can arise when we try to reparametrize a regular path by arc length:

*

*The arc length element $\|\alpha'(t)\|\, dt$ may have no elementary antiderivative.

*Even if the arc length integral can be calculated in closed form, it may be impossible to invert in closed form.

Differential geometry textbooks carefully construct regular paths for which the arc length can be computed, and sometimes inverted. If we write down a "random" regular path, one of these two points is likely to obstruct our ability to "find" a unit-speed path.
Here are some examples where the arc length can be calculated, and (in some cases) explicitly inverted:
$$
\alpha(t) = (t^{2}, t^{3});\quad
\alpha(t) = (t, \cosh t);\quad
\alpha(t) = (t, \sqrt{1 - t^{2}});\quad
\alpha(t) = (t, e^{t}).
$$
