Let $\textit{F}$ be the family of $C^1$ curves in $\mathbb{R}^2$ of fixed length $\bar{l}$ and fixed tangent's turning angle $\bar{k}$. 


What are the curves of positive curvature in $\textit{F}$ minimizing the Euclidean distance between the starting and the ending point? Is the arc of circle of proper radius one of those?


EDIT: as it has been pointed out in the comments, this problem is trivial if curvature is allowed to change sign. What I had in mind and I missed to specify are curves of positive curvature (in the $C^1$ case, curves whose tangent is always turning clockwise).



What curve of positive curvature minimizes distance from the origin, given length and total curvature?
The minimizer is not smooth, it is formed by two sides of an isosceles triangle with angle $\pi-\bar k$ at its vertex. You can approximate it by a smooth curve with large curvature around the vertex and nearly zero curvature elsewhere.


