In many conversations with fellow skiers, I've often heard the phrase "the built in *turn* radius of my skis is x meters". I've always been somewhat puzzled by the implicit message contained here that somehow the ski would have a "default", built-in *turn* radius that corresponds to the *shape* radius given in the specs...?
To my understanding, the given radius for a ski is a measure of the sidecut, i.e. the theoretical arc that the ski manufacturer used to "cut" the ski, but it doesn't, at least directly, correspond to the radii of the turns that you can make with the ski.
As far as I can understand, the way a ski (shaped or not) does a carved turn is by bending; that is, if you want the ski to turn without skidding, you will have to bend it. The sharper a turn you want to make, the more you will have to bend the ski. Even if you have a radically shaped modern ski, with really small "factory" radius, and put it 90 degrees on edge against the snow, if it's completely flat, it will go straight (with maybe minor steering caused by the tip).
So, how come a modern shaped ski is easier to do a carved small radius turn on, than older, straight skis ? My guess is that what the "shapedness" really does, is to allow the ski to be bent with much less force than a straight ski: since a radically shaped ski, when put onto edge, seems to "ride" more on the wide tip & tail, with the waist having less snow pressure against it, it's easier to bend into an arc. The more shape, the more the waist is "off" the snow, and the easier it is to bend the ski to a smaller arc (everything else, e.g ski stiffness, being equal).
Cheers,
Tommy