An official website of the United States government
This gives an improvment of Peter Jones's traveling salesman theorem that holds for Jordan curves, but not for general sets. His theorem implies that the length of a Jordan arc is bounded by (1+delta)diameter + C(delta) beta-sum, and this paper shows this can be replaced by chord + O(beta-sum), where O(diameter) is replaced by the distance between the endpoints of the curve. This is true in all finite dimensions (with a dimension dependent constant). A corollary of our self-contained argument proves the ususal TST in all dimensions (a result of Okikiolu). An appendix proves a folklore result that several different formulation of Jones's theorem are all equivalent.
more »
« less
Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
-
-
For any simple polygon P we compute the optimal upper and lower angle bounds for triangulating P with Steiner points, and show that these bounds can be attained (except in one special case). The sharp angle bounds for an N -gon are computable in time O(N ), even though the number of triangles needed to attain these bounds has no bound in terms of N alone. In general, the sharp upper and lower bounds cannot both be attained by a single triangulation, although this does happen in some cases. Surprisingly, we prove the optimal angle bounds for polygonal triangulations are the same as for triangular dissections. The proof of this verifies, in a stronger form, a 1984 conjecture of Gerver.more » « less
-
For any simple polygon P we compute the optimal upper and lower angle bounds for triangulating P with Steiner points, and show that these bounds can be attained (except in one special case). The sharp angle bounds for an N-gon are computable in time O(N), even though the number of triangles needed to attain these bounds has no bound in terms of N alone. In general, the sharp upper and lower bounds cannot both be attained by a single triangulation, although this does happen in some cases. For example, we show that any polygon with minimal interior angle θ has a triangulation with all angles in the interval I = [θ, 90°–min(36°, θ)/2], and for θ ≤ 36° both bounds are best possible. Surprisingly, we prove the optimal angle bounds for polygonal triangulations are the same as for triangular dissections. The proof of this verifies, in a stronger form, a 1984 conjecture of Gerver.more » « less
Full Text Available