Symmetric polyhedral scenes and parallel redrawings

Source abstract

Liftings and parallel redrawings are classical topics in applied discrete geometry, concerned respectively with vertically lifting a $(d-1)$-picture -- a realisation of a vertex-hyperplane incidence geometry in $\mathbb{R}^{d-1}$ -- to a $d$-dimensional polyhedral scene, and with redrawing a $d$-picture (or hyperplane arrangement) within $\mathbb{R}^d$ while preserving prescribed hyperplane normals. In this paper, we develop a unified framework for the analysis of forced-symmetric liftings and forced-symmetric parallel redrawings. In particular, we show that the classical duality for these theories also appears in the forced-symmetric setting. We establish the orbit lifting matrix and the orbit concurrence geometry matrix, and show that they are the appropriate symmetry-adapted analogues of the standard lifting matrix and concurrence geometry (or parallel redrawing) matrix. We also make explicit the relationship with the orbit version of Whiteley's parallel design matrix for graphs. Using these tools, we derive necessary conditions for symmetric pictures to be forced-symmetric flat, and for symmetric hyperplane arrangements to be forced-symmetric robust, expressed as sparsity counts on the group-labelled quotient graphs associated with the symmetric incidence geometries. Finally, we discuss conjectures regarding the sufficiency of these conditions for generic configurations.