ISSN:
1432-0541
Keywords:
Line weavings
;
Lines in space
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01190155
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