A simplified fold number detector for shapes with monotonic radii

This paper concentrates on the fold number detection problem for the shapes with monotonie radii. The proposed method is extremely simple. Two monotonicity conditions are derived to ensure that the smallest positive integer l making ∫∫_(r,0)∈sr²e^il0dr d0 nonzero is exactly the fold number of the given shape S. The fold numbers of regular polygons, roses, bolt nuts, and other kinds of shapes discussed in the present paper, can therefore be detected quite easily. Note especially that the proposed method uses no matching procedure, a procedure essential in many reported methods. Theoretical properties, mathematical proofs, illustrative figures, and experimental results, are all included in this paper.