Elsevier

Computational Geometry

Volume 36, Issue 1, January 2007, Pages 16-38
Computational Geometry

On the geometric dilation of closed curves, graphs, and point sets

https://doi.org/10.1016/j.comgeo.2005.07.004Get rights and content
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Abstract

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the ratio between the length of a shortest path connecting p and q in G and their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G).

Ebbers-Baumann, Grüne and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δπ/21.57. They conjectured that the lower bound is not tight.

We use new ideas like the halving pair transformation, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to (1+10−11)π/2. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds.

Keywords

Computational geometry
Convex geometry
Convex curves
Dilation
Distortion
Detour
Lower bound
Halving chord
Halving pair
Zindler curves

Cited by (0)

Some results of this article were presented at the 21st European Workshop on Computational Geometry (EWCG '05) [A. Dumitrescu, A. Grüne, G. Rote, Improved lower bound on the geometric dilation of point sets, in: Abstracts 21st European Workshop Comput. Geom., Technische Universiteit Eindhoven, 2005, pp. 37–40], others at the 9th Workshop on Algorithms and Data Structures (WADS '05) [A. Dumitrescu, A. Ebbers-Baumann, A. Grüne, R. Klein, G. Rote, On geometric dilation and halving chords, in: Proc. 9th Workshop on Algorithms and Data Structures (WADS 2005), Lecture Notes Comput. Sci., vol. 3608, Springer, Berlin, August 2005, pp. 244–255].

1

Ansgar Grüne was partially supported by a DAAD PhD-grant.

2

Rolf Klein was partially supported by DFG-grant KL 655/14-1.