Barequet & Rogol (2007) consider the problem of computing the axiality exactly, for both convex and non-convex polygons. The set of all possible reflection symmetry lines in the plane is (by projective duality) a two-dimensional space, which they partition into cells within which the pattern of crossings of the polygon with its reflection is fixed, causing the axiality to vary smoothly within each cell. They thus reduce the problem to a numerical computation within each cell, which they do not solve explicitly. The partition of the plane into cells has O ( n 4 ) cells in the general case, and O ( n 3 ) cells for convex polygons; it can be constructed in an amount of time which is larger than these bounds by a logarithmic factor. Barequet and Rogol claim that in practice the area maximization problem within a single cell can be solved in O ( n ) time, giving (non-rigorous) overall time bounds of O ( n 4 ) for the convex case and O ( n 5 ) for the general case.