There are some differing concepts regarding this cast and how it should be performed. For me the Curve cast is a useful cast when you need to create a uniformed curve of the fly line between the rod tip and the fly, not a sharp elbow at the far end which is a hook cast (hence the name) (see Hook cast) this is for an entirely different application altogether. Curve Cast example:
If you are faced with a situation such as a large section of reeds on a Stillwater for instance which are growing around a natural bend along the bank, (quite common on a stillwater) it would be a great advantage to be able to cast your fly line in a progressive curve around these reeds to target fish that may be feeding along this potentially rich area, thus allowing you to retrieve your fly along the natural curve of the reeds as it will inevitably follow the path of your fly line on it's return. Or:
If you are casting across an almost uniformed current on a river, but the middle section is just slightly faster than the rest, but not fast enough to warrant a reach mend, (see Reach mend cast) then an upstream Curve cast will come in handy. Shooting line into the Curve cast will add extra line to the curve and help to prevent the fly line form straightening out again before touching down.
Always bear in mind that a good Curve cast is created via a deviation in your tracking e.g. deviating from a straight line path usually during the forward stroke. I say usually, as it can also be created in the back cast if required. Curves can be introduced in several ways and can be used in other scenarios too, so why not come along and give them a go.