Lifting symmetric pictures to polyhedral scenes

Scene analysis is concerned with the reconstruction of d-dimensional objects, such as polyhedral surfaces, from (d − 1)-dimensional pictures (i.e., projections of the objects onto a hyperplane). In this paper we study the impact of symmetry on the lifting properties of pictures. We first use methods from group representation theory to show that the lifting matrix of a symmetric picture can be transformed into a block-diagonalized form. Using this result we then derive new symmetry-extended counting conditions for a picture with a non-trivial symmetry group in an arbitrary dimension to be minimally flat (i.e., ‘non-liftable’). These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the various symmetry operations of the picture. We then also transfer lifting results for symmetric pictures from Euclidean (d − 1)-space to Euclidean d-space via the technique of coning. Finally, we offer some conjectures regarding sufficient conditions for a picture realized generically for a symmetry group to be minimally flat

Characterizing minimally flat symmetric hypergraphs

In Kaszanitzky and Schulze (2017) we gave necessary conditions for a symmetric d-picture (i.e., a symmetric realization of an incidence structure in Rd) to be minimally flat, that is, to be non-liftable to a polyhedral scene without having redundant constraints. These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the elements of its symmetry group. In this paper we show that these conditions on the fixed structural components, together with the standard non-symmetric counts, are also sufficient for a plane picture which is generic with three-fold rotational symmetry C3 to be minimally flat. This combinatorial characterization of minimally flat C3-generic pictures is obtained via a new inductive construction scheme for symmetric sparse hypergraphs. We also give a sufficient condition for sharpness of pictures with C3 symmetry

Sufficient Conditions for the Global Rigidity of Periodic Graphs

AbstractTanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension $$d>2$$ d > 2 . We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in $${\mathbb {R}}^d$$ R d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in $${\mathbb {R}}^{2d}$$ R 2 d to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks

Rigid cylindrical frameworks with two coincident points

We develop a rigidity theory for graphs whose vertices are constrained to lie on a cylinder and in which two given vertices are coincident. We apply our result to show that the vertex splitting operation preserves the global rigidity of generic frameworks on the cylinder, whenever it satisfies the necessary condition that the deletion of the edge joining the split vertices preserves generic rigidity

Global Rigidity of Periodic Graphs under Fixed-lattice Representations

In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in Rd. Jackson and Jordán [10] confirmed that these conditions are also sufficient in R2, giving a combinatorial characterization of graphs whose generic realizations in R2 are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic frameworks in R2 in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jordán