phi-Self-Approaching Curves
Oswin Aichholzer, Franz Aurenhammer, Christian Icking, Rolf Klein, Elmar Langetepe, Günter Rote · 1997
We consider all planar oriented curves that have the following property. For each point B on the curve, the rest of the curve lies inside a wedge of angle φ with apex in B; here φ < π is fixed. This property restrains the curve's meandering. 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 1 + cos φ.