An elementary digital plane recognition algorithm

A naive digital plane is a subset of points (x, y, z) ∈ Z3 verifying h ≤ ax+ by+ cz < h+max{|a|, |b|, |c|} where (a, b, c, h) ∈ Z4. Given a finite unstructured subset of Z3, determine whether there exists a naive digital plane containing it is called digital plane recognition. This question is rather classical in the field of digital geometry (also called discrete geometry). We suggest in this paper a new algorithm to solve it. Its asymptotic complexity is bounded by O(n7) but its behavior seems to be linear in practice. It uses an original strategy of optimization in a set of triangular facets (triangles). The code is short and elementary (less than 300 lines) and available on http://www.loria.fr/~debled/plane.