Generalized SelfApproaching Curves # Oswin Aichholzer 1 , Franz Aurenhammer 1 , Christian Icking 2 , Rolf Klein 2 , 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 # # # 2 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. 1 Introduction 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 Collections: Computer Technologies and Information Sciences