Straight crucifix swords are commonly associated with European cultures. By the First Crusade however, curved swords largely supplanted straight swords for military use in contemporary eastern cultures. In East Asia, the dao began replacing the jian even as early as the Han dynasty. Indian and Arabian cultures also adopted curved swords (the talwar and shamshir respectively) comparatively earlier than the Europeans. What factors led to this difference in sword design philosophy?
Never, ever, underestimate the role of fashion in sword design. Long swords were elite weapons. ("Long" here refers literally to the length, and not to any particular style of weapon.) Elite weapons were desirable because they announced your status merely by wearing them, without even fighting. And in martial cultures, you spent a lot more time standing around wearing swords than you actually spent in combat using them. So their social utility was at least as important as their martial utility in terms of understanding why they looked the way they did.
Medieval European sword design inherited from Roman antecedents, and in particular the spatha, which was a design inspired by Celtic long swords. As a long sword, the spatha was expensive and well-suited to cavalry, so it established itself as the continental sword of the aristocracy quite easily, and the aristocratic sidearms of Europe were dominated by long, straight, double-edged blades for many centuries thereafter.
Curved swords existed side-by-side with medieval long swords for most of this time, and they were, in fact, very popular, so it is untrue that European swords in general were in the straight, crucifix style. But curved swords were not considered "elite" weapons. They were short, cheap, practical weapons like falchions, which drew from a long tradition of agricultural or sickle-style weapons that were associated with peasantry and commoners. They were looked down on as status symbols, even when regarded as useful weapons. Even those who could afford horses and fancy long swords would often carry a short curved sword like a falchion with them for when things got down and dirty in the melee.
The Muslim world (as encountered by Europeans) overlapped substantially with the ancient domains of the Roman Empire, and its notions of elite swords were not dissimilar. During the Crusades, Arab swords were typically long, straight, double-edged, and single handed. They had a bit less of the crucifix-style hilt going on, but that's mostly just the guard design, and that didn't come from the Romans, anyway. So there really wasn't much in the way of "international influence" to steer western sword design away from the straight long sword.
Until the Turks. Although curved "sickle swords" were known throughout the world, their roots in agricultural labour didn't give them a lot of social cachet. The Turks may have been the first to lengthen the curved sword into an elite cavalry weapon, somewhere around the 8th Century. (There is also a short Turkish curved sword called a yataghan, but it did not have nearly so much influence.) But even so, the peoples of the Asian steppes were treated as barbarians by most of the cultures and empires who encountered them, so the mere existence of a long, curved sword that was useful from horseback would not have been enough to convince western aristocrats that it was a proper badge of status.
But after the Mongols and then the Turks overran much of western Asia and established their own empires, perceptions about elite weapons began to shift. What does "elite" mean, after all, other than "associated with the ruling class"? After Turks have been running the show for a while, their weapon styles began to redefine the general conception of "aristocratic cool". (And kicking some highly respected ass all through the region certainly didn't hurt the reputation of their weaponry, either.) This happened earliest in lands like Persia and India (giving us weapons like the scimitar and talwar), but it eventually came to Europe as well, with Ottoman incursions into, and rule over, eastern Europe in the 14th through 19th Centuries.
This led to the European take on the Turkish sword, which is generally known as the sabre, and is so solidly ingrained into our own cultural patterns that we hardly recognize its Turkish influences anymore. That's partly because the sword was popularized in Western Europe by Hussar regiments, which were modelled after the forces that drove the Turks out of Eastern Europe. Part of its popularity was that it was perceived as the weapon that defeated the Turks, when in fact the opposite was closer to the truth.
So fashionable was the new curved sabre that it became the dominant style of western military sword during the 19th Century, and although the classic straight-bladed, doubled-edged sword (by then called a broadsword) did manage to survive, it mostly did so by rebranding itself as a "sabre". (See, for example, the Patton saber, and the 1908 trooper sabre, both of which are basically rapiers masquerading as broadswords, while calling themselves sabres.)
(This post covers a lot of ground, but Swords and Hilt Weapons is a good overview of the broad historical development of the sword, with Barbarians and Christians and 17th Century Europe, both by Anthony North, covering the main influences on western sword shapes.)
It is not true that "by the First Crusade however, curved swords largely supplanted straight swords for military use in contemporary eastern cultures." The narrow curved sabres seem to have been invented by Turkic-speakers in the Black Sea steppes around the 8th century CE while Arabs and Persians used straight two-edged swords. For many centuries thereafter, Arabs Syrians and Persians associated curved swords with Turks. During the First crusade, most sedentary warriors in all the armies used straight two-edged swords. We can see this by reading Arabic treatises on swords which describe Rus and Frankish sword-blades as similar in shape to Yemeni or Egyptian or Kharwazemi sword-blades. The classic Persian shamshir was first invented in the 15th century CE.
Around the year 1000, big fighting knives stop appearing in the ground or artwork in Catholic countries, and swords forged in those countries get more cross-shaped. Asymetric weapons start to reappear in France and its neighbours from the 13th century onwards, but they are often showed in the hands of ruffians, rarely preserved in collections, and marked with protective signs to ward off bad influences which don't appear on straight two-edged weapons from the same places. Meanwhile Turks and Mongols made military progress across Eurasia under people like the Sejluks and Genghis Khan and the Mamelukes and the Timurids and the Mughals. In the 16th, 17th, and 18th centuries, most of the Moslems and Eurasian Pagans who Catholics and Protestants met were using curved single-edged swords inspired by Turkish and Mongol fashions. But at the same time, Europeans were losing the taboo against single-edged asymmetrical weapons, until in the 19th century most European military swords are single-edged and curved.
So in the end, all it is is a stereotype build out of former Christian taboos and accidents of history. You could just as well argue that in the early middle ages, most of the peoples from Iran to Iberia used straight two-edged swords, and in the last thousand years many of them switched to curved single-edged swords.
Further reading: David Nicolle, Medieval Warfare Sourcebook (2 vols)
Manouchehr Khorasani, Arms and Armour from Iran
James G. Elmslie's messer and falchion typology
By the First Crusade however, curved swords largely supplanted straight swords for military use in contemporary eastern cultures.
At the time of the First Crusade, curved swords were used in Central Asia (the early steppe sabre), and were seeing some use in China, and had recently come into use in Japan. In Western Asia, the double-edge straight sword was still supreme, and was used by both sides in the Crusades.
The spread of curved single-edged swords began in Central Asia in about the 7th century. Early of single-edged steppe swords included straight examples and forward-curved examples as well as "normal" backward-curved swords. These spread widely across the steppe, and appeared in Europe by the 10th century in the hands of steppe peoples (e.g., Cumans). One famous European example from this time is the "Sabre of Charlemagne":
The Song Dynasty saw a mix of straight and curved dao being used - early dao from the Han through to the Tang were usually straight. The increasing use of curved dao in China is usually attributed to Turko-Mongol influence, from the Mongols joining the Jin and Song and Xi Xia in their wars for the control of China. The Japanese switch from straight tachi to curved tachi appears to have begun earlier, and might be an entirely dependent development.
Elsewhere, the spread of the curved sabre appears to be closely linked to the spread of Turkic peoples, and their military employment. The curved sword became usual in Persia in the 15th century, and common in Eastern Europe and India in the 16th century (with straight swords continuing in use in both Eastern Europe and India alongside curved swords). Curved swords proceeded to spread across Europe from east to west, with the start of the 19th century seeing a mix of curved and straight swords in military use.
Part of the 18th century Western European shift to curved swords is due to cavalry becoming the major sword users. Infantry were the dominant group on the battlefield, but when the only infantry wearing swords are officers, and cavalry - troopers and officers alike - wear swords, the cavalry sword is the common sword, especially in terms of actual use as a weapon on the battlefield.
In at least some places (e.g., 17th century Europe), the adoption of the sabre in place of the straight sword coincides with a reduction in armour, and increased use of firearms. Together, these mean that there was less need for a sword point to find gaps in armour, which might have favoured sabres. However, in other places/times, there doesn't appear to be any connection with armour or firearms, so it isn't anything as simple as "guns = less armour and fewer infantry swords = cavalry using curved swords".
Finally, it should be noted that many curved swords had very little curvature - not enough to make any significant functional difference. For example, consider these late 19th century Qing cavalry swords:
For more discussion, and references and many examples of swords, see my past answers in
