Summary: Generalized SelfApproaching Curves #
Oswin Aichholzer 1 , Franz Aurenhammer 1 , Christian Icking 2 , Rolf Klein 2 ,
Elmar Langetepe 2 , and G˜unter Rote 1
1 Technische Universit˜at Graz, A8010 Graz, Austria.
2 FernUniversit˜at Hagen, Praktische Informatik VI, D58084 Hagen, Germany.
Abstract. We consider all planar oriented curves that have the follow
ing property depending on a fixed angle #. For each point B on the
curve, the rest of the curve lies inside a wedge of angle # with apex
in B. 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
c(#) 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 #.
Keywords. Selfapproaching curves, convex hull, detour, arc length.
Let f be an oriented curve in the plane running from A to Z, and let # be an
angle in [0, #). Suppose that, for every point B on f , the curve segment from