Oswin Aichholzer, Franz Aurenhammer, Christian Icking, Rolf Klein, Elmar
Langetepe, and Günter Rote:
Generalized self-approaching curves
In: "Algorithms and Computation—Ninth Annual International
Symposium on Algorithms and Computation. Taejon, Korea, December
1998". Proceedings of ISAAC'98. Editors: Kyung-Yong Chwa and
Oscar H. Ibarra. Lecture Notes in Computer Science 1533,
Springer-Verlag, 1998, pp. 317–327. doi:10.1007/3-540-49381-6_34
Discrete Applied Mathematics 109 (2001), 3–24. doi:10.1016/S0166-218X(00)00233-X
For an angle φ between 0 and 180o, we consider the class of
oriented curves which are φ-self-approaching in the following sense:
for any point A on the considered curve, the rest of the curve is
inside a wedge of angle φ at A. This is a direct generalization
of self-approaching curves which are 90o-self-approaching. We
prove a tight upper bound on the length of a φ-self-approaching curve
in terms of the distance between its endpoints. The upper bound only depends
on the angle φ.
This problem arises in the performance analysis of certain on-line navigation
strategies. A closely related problem concerns curves
with increasing chords.
Last update: April 5, 2002.