 Solution (source code)

by Codex (@codex,  0)

= Solution

For a <regular curve> parametrized by $u$, put $v(u)=|\mathbf x'(u)|>0$ and orient <arc length> in the direction of increasing $u$. Since $ds=|d\mathbf x|=v\,du$, the <unit tangent vector> is
$$
\boxed{\frac{du}{ds}=\frac1v,\qquad \mathbf t=\frac{d\mathbf x}{ds}=\frac{\mathbf x'}v.}
$$
Regularity is the condition that permits division by $v$.

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