Gift-wrapping based preimage computation algorithm

Based on a classical convex hull algorithm called gift-wrapping, the purpose of the paper is to provide a new algorithm for computing the vertices of a polytope called preimage—roughly the set of naive digital planes containing a finite subset SS of Z3Z3. The vertices of the upper hemisphere, the ones of the lower hemisphere and at last the equatorial vertices are computed independently. The principle of the algorithm is based on duality and especially on the fact that the vertices of the preimage correspond to faces of the input set SS or of its chords set S⊖S∪{(0,0,1)}S⊖S∪{(0,0,1)}. It allows to go from one vertex to another by gift-wrapping until the whole region of interest has been explored.