Abstract
Aweaving W is a simple arrangement of lines (or line segments) in the plane together with a binary relation specifying which line is “above” the other. A system of lines (or line segments) in 3-space is called arealization ofW, if its projection into the plane isW and the “above-below” relations between the lines respect the specifications. Two weavings are equivalent if the underlying arrangements of lines are combinatorially equivalent and the “above-below” relations are the same. An equivalence class of weavings is said to be aweaving pattern. A weaving pattern isrealizable if at least one element of the equivalence class has a three-dimensional realization. A weaving (pattern)W is calledperfect if, along each line (line segment) ofW, the lines intersecting it are alternately “above” and “below.” We prove that (i) a perfect weaving pattern ofn lines is realizable if and only ifn ≤ 3, (ii) a perfect m byn weaving pattern of line segments (in a grid-like fashion) is realizable if and only if min(m, n) ≤ 3, (iii) ifn is sufficiently large, then almost all weaving patterns ofn lines are nonrealizable.
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Communicated by Takao Asano.
Jànos Pach has been supported in part by Hungarian NFSR Grant 1812, NSF Grant CCR-8901484, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center, under NSF Grant STC88-09648. Richard Pollack has been supported in part by NSA Grant MDA904-89-H-2030, NSF Grants DMS-85-01947 and CCR-8901484, and DIMACS. Emo Welzl has been supported in part by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM) and DIMACS.
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Pach, J., Pollack, R. & Welzl, E. Weaving patterns of lines and line segments in space. Algorithmica 9, 561–571 (1993). https://doi.org/10.1007/BF01190155
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DOI: https://doi.org/10.1007/BF01190155