  Choose two points in the tangent bundle of the Euclidean plane
$(x,X),(y,Y)\in T{ \mathbb R}^2$. In this work we characterise the immersed
length minimising paths with a prescribed bound on the curvature starting at
$x$, tangent to $X$; finishing at $y$, tangent to $Y$, in each connected
component of the space of paths with a prescribed bound on the curvature from
$(x,X)$ to $(y,Y)$.
