Generalized self-approaching curves
- Aichholzer, Oswin
- Aurenhammer, Franz
- Icking, Christian
- Klein, Rolf
- Langetepe, Elmar
- Rote, Günter
Publication date
April 2001
DOI Publisher
Elsevier Science B.V. ISSN
0166-218XAbstract
AbstractWe consider all planar oriented curves that have the following property depending on a fixed angle ϕ. For each point B on the curve, the rest of the curve lies inside a wedge of angle ϕ with apex in B. This property restrains the curve's meandering, and for ϕ⩽π/2 this means that a point running along the curve always gets closer to all points on the remaining part. For all ϕ<π, we provide an upper bound c(ϕ) 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+cosϕ
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