Consider two orthogonal closed chains on a cylinder. These chains are monotone with respect to the tangential Θ direction. We wish to rigidly move one chain so that the total area between the two is minimized. This minimization is a geometric measure of similarity between two melodies proposed byÓ Maidín. The Θ direction represents time and the axial direction, z, represents pitch. Let the two chains have n and m vertices respectively, where n ≥ m, We present an O(n + m) time algorithm if Θ is

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... ixed, and an O(nm log(n + m)) time algorithm for general rigid motions. These bounds also apply for planar orthogonal monotone open chains, where area is measured only within the common domain of the two chains in the direction * This paper extends the results presented by the authors at CCCG'03 [1]