In addition to everything Martin said about the use of curved swords, the curve also made reading the blade's arc difficult. It was hard to tell where the sword blade would end up. The curve also added strength to the blade. When using a straight, double-edged sword, the weilder would have to remember to block using the flat of the blade. Otherwise, at best, the blade would be notched, and at worst, it could result in a shattered blade (for either combatant). The Japanese used a slightly curved blade almost to the exclusion of other styles of blades, but these were single-edged weapons that the weilder would have to be trained (extensively) to use.

-RobRoy