The Tachi was really the Japanese sabre of the armour period which was usually carried cutting edge downwards, typically had more length & curvature and more taper and a cutting edge designed to penetrate armour. It is true that was rather down the priority list of weapons but was also wielded from horseback.
The evolution into the Katana, being one of the two part Daisho set, happened in the transition to more 'peaceful era' (if we can call it that). These were carried (and more significantly stored) edge up. This period is the phase were much of the fencing was refined to a Zen activity and also the blades began to be forged and polished as an artform in themselves.
