Name
Abstract | ||
|---|---|---|
We study non-crossing frameworks in the plane for which the clas- sical reciprocal on the dual graph is also non-crossing. We give a complete de- scription of the self-stresses on non-crossing frameworks G whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the stress on G; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigid- ity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed ver- tex. We show that for such pseudo-triangulation embeddings of planar Laman circuits which are suciently generic, the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation Laman circuits. All self-stresses on a planar graph correspond to liftings to piece-wise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs. |
Year | DOI | Venue |
|---|---|---|
2004 | 10.1007/s00454-004-1139-x | Discrete & Computational Geometry |
Keywords | Field | DocType |
Computational Mathematic,Special Interest,Geometric Property,Planar Graph,Sign Pattern | Topology,Reciprocal,Combinatorics,Laman graph,Embedding,Vertex (geometry),Planar straight-line graph,Dual graph,Piecewise linear function,Mathematics,Planar graph | Journal |
Volume | Issue | ISSN |
32 | 4 | Discrete Comput. Geom., 32:4 (2004), 567-600 |
Citations | PageRank | References |
4 | 0.58 | 10 |
Authors | ||
6 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| David Orden | 1 | 160 | 20.26 |
| Günter Rote | 2 | 1181 | 129.29 |
| Francisco Santos | 3 | 184 | 18.73 |
| Brigitte Servatius | 4 | 141 | 19.37 |
| Herman Servatius | 5 | 99 | 14.35 |
| Walter Whiteley | 6 | 450 | 32.34 |