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#SelfApproaching Curves (extended abstract)

Summary: #­Self­Approaching Curves
(extended abstract)
Oswin Aichholzer ## Franz Aurenhammer ## Christian Icking #
Rolf Klein # Elmar Langetepe # G˜unter Rote ##
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


Source: Aurenhammer, Franz - Institute for Theoretical Computer Science, Technische Universität Graz
Technische Universität Graz, Institute for Software Technology


Collections: Computer Technologies and Information Sciences