I want to find a function $\gamma(t) = (x(t),y(t))$ such that for two values of $t$, we have $\gamma'(t)$ and $\gamma(t)$ have some value, and at no point does the curvature ever exceed $r$.
What sort of techniques can I use to find such a function?
I know how to this for one value of $t$ (it would just be a line with the desired slope and intercept), but I don't know how to combine the results for both values of $t$.
How can I approach this problem?
EDIT: If it helps, the known values of $\gamma$ (and $\gamma'$) are real numbers, not dependent on any variables.