Weaving patterns of lines and line segments in space

Abstract

A weaving 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. An m by n bipartite weaving consists of m horizontal and n vertical lines or line segments which mutually intersect. A system of lines (or line segments) in 3-space is called a realization of W, if its projection into the plane is W and the relative positions of the lines respect the “above” specifications. An equivalence class of weavings induced by the combinatorial equivalence of the underlying planar arrangement of lines is said to be a weaving pattern. A weaving pattern is realizable if at least one element of the equivalence class has a realization. A weaving (pattern) W is called perfect, if along each line of W, the lines intersecting it are alternately “above” and “below”. We prove that (i) a perfect weaving pattern of n lines is realizable if and only if n≤3, (ii) a perfect m by n bipartite weaving pattern is realizable if and only if min(m, n)≤3, (iii) if n is sufficiently large then almost all weaving patterns of n lines are nonrealizable.KeywordsLine SegmentLine DiagramPlanar ArrangementCourant InstituteSimple ArrangementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.