Oswin Aichholzer, Franz Aurenhammer, Christian Icking, Rolf Klein, Elmar
Langetepe, and Günter Rote:
Generalized selfapproaching curves

In: "Algorithms and Computation—Ninth Annual International
Symposium on Algorithms and Computation. Taejon, Korea, December
1998". Proceedings of ISAAC'98. Editors: KyungYong Chwa and
Oscar H. Ibarra. Lecture Notes in Computer Science 1533,
SpringerVerlag, 1998, pp. 317–327. doi:10.1007/3540493816_34

Discrete Applied Mathematics 109 (2001), 3–24. doi:10.1016/S0166218X(00)00233X
Abstract
For an angle φ between 0 and 180^{o}, we consider the class of
oriented curves which are φselfapproaching 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 selfapproaching curves which are 90^{o}selfapproaching. We
prove a tight upper bound on the length of a φselfapproaching 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 online navigation
strategies. A closely related problem concerns curves
with increasing chords.
Last update: April 5, 2002.