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We study the problem of how to obtain an integer realization of a 3d polytopewhen an integer realization of its dual polytope is given. We focus on gridembeddings with small coordinates and develop novel techniques based on Colinde Verdi\`ere matrices and the Maxwell-Cremona lifting method. We show thatevery truncated 3d polytope with n vertices can be realized on a grid of sizeO(n^{9log(6)+1}). Moreover, for every simplicial 3d polytope with n verticeswith maximal vertex degree {\Delta} and vertices placed on an L x L x L grid, adual polytope can be realized on an integer grid of size O(n L^{3\Delta + 9}).This implies that for a class C of simplicial 3d polytopes with bounded vertexdegree and polynomial size grid embedding, the dual polytopes of C can berealized on a polynomial size grid as well.

A Duality Transform for Constructing Small Grid Embeddings of 3d Polytopes