# Generalized self-approaching curves

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

### Abstract:

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 .

Reference: O. Aichholzer, F. Aurenhammer, C. Icking, R. Klein, E. Langetepe, and G. Rote. Generalized self-approaching curves. Discrete Applied Mathematics, 109(1-2):3-24, 2001. Special Issue. [SFB-Report F003-134, TU Graz, Austria, 1998].