Length minimising bounded curvature paths in homotopy classes

Abstract Choose two points in the tangent bundle of the Euclidean plane ( x , X ) , ( y , Y ) ∈ T 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 ) .