In Prof Ted Shifrin's differential geometry book linked here, problem 22 on page 21 is about obtaining an integral representation for a curve $\boldsymbol{\alpha}(s)$ of constant torsion $c$ using the binormal vector $\boldsymbol{b}$ defined over an interval $[a,b]$. This is of the form
$$ \boldsymbol{\alpha}(t) = \frac{1}{c} \int_a^t \boldsymbol{b}(u) \times \boldsymbol{\dot{b}}(u)\ \mathrm{d}u,$$
with $t \in [a,b]$.
I am trying to obtain a similar representation for a curve of constant curvature. I think this should be possible, But I am not sure how to go about this problem? Any hints?? I apologise if this is something trivial and I am missing something simple.