
Notice they were carried blade down
The Samurai as we think of them originally developed as horse soldiers, cavalry. Those who fought on foot, ashigaru, can technically be classified as members of the Samurai class but that's a discussion for another day. Now, the katana (originally called uchigatana) didn't come into existence until just about the 15th century (circa the 1390s specifically). It's predecessor, the tachi, was effectively a cavalry saber and was developed sometime between 900 and 1100AD. Curved blades were far more efficient than the straight swords of the ashigaru when wielded on horseback, where the curve of the blade adds considerably to the downward force of the cutting motion. Therefore, on a cavalry dominated battlefield, it only made sense to have a significantly curved blade (the tachi also had a proportionally long handle to counterbalance the sword for one-handed use).