#SelfApproaching Curves (extended abstract) Summary: #­Self­Approaching Curves (extended abstract) Oswin Aichholzer ## Franz Aurenhammer ## Christian Icking # Rolf Klein # Elmar Langetepe # G˜unter Rote ## Abstract 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 # with apex in B; here # < # 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 #. 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 B to Z is contained in a wedge of angle # with apex in B. Then the curve f is called #­self­ approaching. The name refers to the fact that, at least for # # #/2, the curve always gets closer to each point on its remaining part. The quantity length(f) Collections: Computer Technologies and Information Sciences