English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study non-crossing frameworks in the plane for which the classicalreciprocal on the dual graph is also non-crossing. We give a completedescription of the self-stresses on non-crossing frameworks whose reciprocalsare non-crossing, in terms of: the types of faces (only pseudo-triangles andpseudo-quadrangles are allowed); the sign patterns in the self-stress; and ageometric condition on the stress vectors at some of the vertices.  As in other recent papers where the interplay of non-crossingness andrigidity of straight-line plane graphs is studied, pseudo-triangulations showup as objects of special interest. For example, it is known that all planarLaman circuits can be embedded as a pseudo-triangulation with one non-pointedvertex. We show that if such an embedding is sufficiently generic, then thereciprocal is non-crossing and again a pseudo-triangulation embedding of aplanar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulationembedding of a planar Laman circuit, the reciprocal is still non-crossing and apseudo-triangulation, but its underlying graph may not be a Laman circuit.Moreover, all the pseudo-triangulations which admit a non-crossing reciprocalarise as the reciprocals of such, possibly singular, stresses onpseudo-triangulation embeddings of Laman circuits.  All self-stresses on a planar graph correspond to liftings to piece-wiselinear surfaces in 3-space. We prove characteristic geometric properties of thelifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure

Non-Crossing Frameworks with Non-Crossing Reciprocals