A duality transform for constructing small grid embeddings of 3d polytopes

We study the problem of how to obtain an integer realization of a 3d polytope when an integer realization of its dual polytope is given. We focus on grid embeddings with small coordinates and develop novel techniques based on Colin de Verdière matrices and the Maxwell–Cremona lifting method.We show that every truncated 3d polytope with n   vertices can be realized on a grid of size O(n9log⁡6+1)O(n9log⁡6+1). Moreover, for every simplicial 3d polytope with n   vertices with maximal vertex degree Δ and vertices placed on an L×L×LL×L×L grid, a dual polytope can be realized on an integer grid of size O(nL3Δ+9)O(nL3Δ+9). This implies that for a class CC of simplicial 3d polytopes with bounded vertex degree and polynomial size grid embedding, the dual polytopes of CC can be realized on a polynomial size grid as well.