Mobility of geometric constraint systems with extrusion symmetry

Owen, John and Schulze, Bernd (2025) Mobility of geometric constraint systems with extrusion symmetry. Contributions to Algebra and Geometry, 66 (2). pp. 345-389. ISSN 0138-4821

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Abstract

If we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector τ, and finally join corresponding points of the two copies, then we obtain a framework with ‘extrusion’ symmetry in the direction of τ. This process may be repeated t times to obtain a framework whose underlying graph has Z2t 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.