phi-Self-Approaching Curves

Aichholzer, Oswin; Aurenhammer, Franz; Icking, Christian; Klein, Rolf GND; Langetepe, Elmar GND; Rote, Günter

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.

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Aichholzer, Oswin / Aurenhammer, Franz / Icking, Christian / et al: phi-Self-Approaching Curves. Hagen 1997. FernUniversität in Hagen.

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