**O. Aichholzer, F. Aurenhammer, C. Icking, R. Klein, E. Langetepe, and
G. Rote**

We consider all planar oriented curves that have the following property
depending on a fixed angle . For each point on the curve, the
rest of the curve lies inside a wedge of angle with apex in .
This property restrains the curve's meandering, and for
this means that a point running along the curve always gets closer to all
points on the remaining part. For all , we provide an upper
bound for the length of such a curve, divided by the distance
between its endpoints, and prove this bound to be tight. A main step is in
proving that the curve's length cannot exceed the perimeter of its convex
hull, divided by
.