ISAAC 1998: Algorithms and Computation pp 317-327

# Generalized Self-Approaching Curves

• Oswin Aichholzer
• Franz Aurenhammer
• Christian Icking
• Rolf Klein
• Elmar Langetepe
• Günter Rote
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1533)

## Abstract

We consider all planar oriented curves that have the following property depending on a fixed angle ϕ. For each point B on the curve, the rest of the curve lies inside a wedge of angle ϕ with apex in B. This property restrains the curve’s meandering, and for ϕ < - π/2 this means that a point running along the curve always gets closer to all points on the remaining part. For all ϕ < π, we provide an upper bound c(ϕ) 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 ϕ.

## Keywords

Self-approaching curves convex hull detour arc length

## References

1. 1.
O. Aichholzer, F. Aurenhammer, C. Icking, R. Klein, E. Langetepe, and G. Rote. ϕ-self-approaching curves. Technical Report 226, Department of Computer Science, FernUniversität Hagen, Germany, 1997. Submitted for publication.Google Scholar
2. 2.
H. Alt, B. Chazelle, and R. Seidel, editors. Computational Geometry. Dagstuhl-Seminar-Report 109. Internat. Begegnungs-und Forschungszentrum für Informatik, Schloss Dagstuhl, Germany, March 1995.Google Scholar
3. 3.
S. Arya, G. Das, D. M. Mount, J. S. Salowe, and M. Smid. Euclidean spanners: short, thin, and lanky. In Proc. 27th Annu. ACM Sympos. Theory Comput., pages 489–498, 1995.Google Scholar
4. 4.
H. P. Croft, K. J. Falconer, and R. K. Guy. Unsolved Problems in Geometry. Springer-Verlag, 1990.Google Scholar
5. 5.
C. Icking and R. Klein. Searching for the kernel of a polygon: A competitive strategy. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 258–266, 1995.Google Scholar
6. 6.
C. Icking, R. Klein, and E. Langetepe. Self-approaching curves. Technical Report 217, Department of Computer Science, FernUniversität Hagen, Germany, 1997. To appear in Mathematical Proceedings of the Cambridge Philosophical Society.Google Scholar
7. 7.
J.-H. Lee and K.-Y. Chwa. Tight analysis of a self-approaching strategy for online kernel-search problem. Technical report, Department of Computer Science, KAIST, Taejon, Korea, 1998Google Scholar
8. 8.
J.-H. Lee, C.-S. Shin, J.-H. Kim, S. Y. Shin, and K.-Y. Chwa. New competitive strategies for searching in unknown star-shaped polygons. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 427–429, 1997.Google Scholar
9. 9.
A. López-Ortiz and S. Schuierer. Position-independent near optimal searching and on-line recognition in star polygons. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 445–447, 1997.Google Scholar
10. 10.
G. Rote. Curves with increasing chords. Mathematical Proceedings of the Cambridge Philosophical Society, 115(1):1–12, 1994.
11. 11.
J. Ruppert and R. Seidel. Approximating the d-dimensional complete Euclidean graph. In Proc. 3rd Canad. Conf. Comput. Geom., pages 207–210, 1991.Google Scholar

## Authors and Affiliations

• Oswin Aichholzer
• 1
• Franz Aurenhammer
• 1
• Christian Icking
• 2
• Rolf Klein
• 2
• Elmar Langetepe
• 2
• Günter Rote
• 1
1. 1.Technische Universität GrazGrazAustria
2. 2.FernUniversität HagenHagenGermany