Curvature-bounded traversals of narrow corridor

bounded curvature paths traversing constant-width regions of the
plane, called corridors. We make explicit a width threshold $\tau$
with the property that (a) all corridors of width at least $\tau$
admit a unit-curvature traversal and (b) for any width $w<\tau$
there exist corridors of width $w$ with no such traversal.
Applications to the design of short, but not necessarily shortest,
and high clearance, but not necessarily maximum clearance,
curvature-bounded paths in general polygonal domains, are also
discussed.