  A bounded curvature path is a continuously differentiable piece-wise $C^2$
path with bounded absolute curvature connecting two points in the tangent
bundle of a surface. These paths have been widely considered in computer
science and engineering since the bound on curvature models the trajectory of
the motion of robots under turning circle constraints. Analyzing global
properties of spaces of bounded curvature paths is not a simple matter since
the length variation between length minimizers of arbitrary close endpoints or
directions is in many cases discontinuous. In this note, we develop a simple
technology allowing us to partition the space of spaces of bounded curvature
paths into one-parameter families. These families of spaces are classified in
terms of the type of connected components their elements have (homotopy
classes, isotopy classes, or isolated points) as we vary a parameter defined in
the reals. Consequently, we answer a question raised by Dubins (Pac J Math
11(2), 1961).
