## David Orden, Günter Rote, Francisco Santos, Brigitte Servatius, Herman
Servatius, and Walter Whiteley:

# Non-crossing frameworks with non-crossing reciprocals

*Discrete and Computational Geometry* **32** (2004), 567–600.
(Special issue in honor of Lou Billera), doi:10.1007/s00454-004-1139-x*,
arXiv:math/0309156 [math.MG].
*
### Abstract

We study non-crossing frameworks in the plane for which the classical
reciprocal on the dual graph is also non-crossing. We give a complete
description 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 rigidity
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 vertex.
We show that for such pseudo-triangulation embeddings of planar Laman circuits
which are sufficiently 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.

Last update: August 15, 2017.