Rigid frameworks with dilation constraints

Abstract

We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean $d$-space under additional dilation constraints in specified coordinate directions. In this setting we obtain a complete characterisation of generic rigidity. We then consider generic global rigidity. In particular, we provide an algebraic sufficient condition and a weak necessary condition. We also construct a large family of globally rigid frameworks and conjecture a combinatorial characterisation when most coordinate directions have dilation constraints.

Figures (1)

• Figure 1: The 2-dimensional framework described in \ref{['ex:c4']} is depicted on the left. This framework is not 2-rigid since there is a non-trivial continuous deformation taking it to the framework on the right. Nevertheless the framework is $(2,1)$-rigid since the dilation constraints in the $y$-coordinates prevent any nontrivial motion. The intuition behind this is to first note that translation in the $y$-direction and rotation evidently break the dilation constraints. Consider now the nontrivial motion depicted. The top left vertex follows the path $\theta \mapsto (1 + \sin \theta, 1 + \cos \theta)$ and the top right vertex follows the path $\theta \mapsto (2 + \sin \theta, 1 + \cos \theta)$. As the bottom two vertices have $y$-coordinate 1, the dilation constraints require that the $y$-coordinates of the top two vertices -- both of which are $1 + \cos \theta$ -- are constant during the motion, a clear contradiction.

Theorems & Definitions (40)

• Theorem 2.1: Asimow and Roth asi-rot
• Lemma 2.2: Maxwell Max
• Lemma 3.1
• proof
• Example 3.2
• Example 3.3
• Example 4.1
• Theorem 4.2
• proof
• Theorem 4.3