Mobility of geometric constraint systems with extrusion symmetry

Abstract

AbstractIf we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector $$\tau $$ τ , and finally join corresponding points of the two copies, then we obtain a framework with ‘extrusion’ symmetry in the direction of $$\tau $$ τ . This process may be repeated t times to obtain a framework whose underlying graph has $$\mathbb {Z}_2^t$$ Z 2 t as a subgroup of its automorphism group and which has ‘t-fold extrusion’ symmetry. Extruding a framework is a widely used technique in CAD for generating a 3D model from an initial 2D sketch, and hence it is important to understand the flexibility of extrusion-symmetric frameworks. Using group representation theory, we show that while t-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with t-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. This allows us to establish Fowler-Guest-type character counts to analyse the mobility of such frameworks. We show that this entire theory also extends to the more general point-hyperplane frameworks with t-fold extrusion symmetry. Moreover, we show that under suitable regularity conditions the infinitesimal flexes we detect with our symmetry-adapted counts extend to finite (continuous) motions. Finally, we establish an algorithm that checks for finite motions via linearly displacing framework points along velocity vectors of infinitesimal motions.