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Cutting vs Thrusting
Most fighting swords compromised between cutting and thrusting, and there are tons of ways to make a blade so that it can both cut and thrust. There are so many variations in sword shape that it's impossible to talk about them all here, but generally speaking blades curve in different ways to create different advantages, and to better balance the weapon or distribute its weight into a given area. As an example, some swords were designed to have a curved edge with a straight back, so that it had the cutting edge of a saber, but also a thrusting point.
There were, of course, those that specialized in one or the other, but who uses what is largely a matter of the needs, preferences, the style of the person wielding it, and the people against whom they most often fight. Even within a given culture and time period, there are often different styles of sword used by different people, just because the different weapons suit the different people. A weapon that can do both cutting and thrusting effectively is, obviously, more flexible than a weapon that can only do one, no matter which one it can do. If your soldiers need to be versatile they're more likely to have a sword that both cuts and thrusts. If they're dedicated to a single style of combat--mounted combat, for instance--or if individual types of troops dedicate themselves to specific techniques--i.e. this soldier is heavy cavalry and those are the tactics they know--you're more likely to get specialized weapons meant for that style.
A great example of a specialized weapon is the two-handed great sword, which was meant for the very specific purpose of cutting through, and defending against, pikemen. That's what it does. The length of the sword gives it reach, the design of the sword (specifically the ricasso, which is a flat, dull bit of metal on the blade side of the crossguard meant to allow half-swording) makes it possible to wield it somewhat like a pike or spear, and the thrusting point allows for good stabbing action. Yet, even with this specialized weapon, it can and would both cut and thrust.
This brings me to the topic of mounted combat because, generally speaking, thrusting from a horse is harder than slashing from a horse. That's just because when thrusting you have a smaller surface area that will do damage and it requires very good aim to hit the target where you want to. Plus, if you add the momentum of the horse into the equation, a slash can be more effective and not require the same precise aim that a thrust might need. There's more cutting area than thrusting area to a blade. A thrust goes deeper and is harder to treat than a slash, so it is more lethal
Another factor to take into account is the type of armor worn by the opponent. Thrusting into a lightly or unarmored opponent can also lead to getting your sword trapped and, if you're mounted, ripped out of your hand because it's buried in somebody's body. The same can be true of chain armors. Thrust the wrong sword into chain armor, and you'll have a hell of a time pulling it back out. Chain can trap blades that do manage to make it through the small gaps in the rings, and blades aren't going to cut through the rings.
However, you're unlikely to penetrate plate armor with either a thrust or a slash, that's kind of the point of plate armor. So if your thrust isn't going to penetrate the armor, and you're basically hoping to knock your opponent around, or get them off their horse, a slash is often your best bet. Making sure you hit the opponent and deliver enough force to give them a concussion, knock them to the ground, or batter them inside their armor is more important than attempting to pierce what will not be pierced. A thrust can do a similar job, but there's a reason lances were invented for thrusting into a mounted target.
Curved swords were preferred by most cavalry and mounted soldiers simply because the curve creates more slashing edge for cutting, which is easier to aim than a thrusting sword. This doesn't mean that all curved swords are meant solely for cutting. As I've said, most swords were designed to give some sort of balance between the two techniques. It's just more flexible that way.
In discussing these things with other writers, I've occasionally run into the perception that curved swords are "eastern," or create the impression of a more "eastern" culture. This is a ridiculous preconception. There are a variety of curved European swords (including the falchion, malchus, storta and messer) and many straight-edged swords developed by non-European cultures. The design of the sword is about what it's meant to do, not where it comes from. Beyond that, who cares? No, seriously, you're creating a culture here. If it makes more sense for them to use a curved sword (for instance if they often fight mounted) then that's what they should use, regardless.