  A bounded curvature path is a continuously differentiable piecewise $C^2$
path with bounded absolute curvature that connects two points in the tangent
bundle of a surface. In this note we give necessary and sufficient conditions
for two bounded curvature paths, defined in the Euclidean plane, to be in the
same connected component while keeping the curvature bounded at every stage of
the deformation. Following our previous results here we finish a program
started by Lester Dubins in 1961.
