Two more cents from someone who does bothI came here via the same link the Jack M followed - from a forum where a bunch of carve-happy snowboarders hang out. I think Jack summed it up pretty well, but I wanted to add a couple things.
First, let me define carving, at least for the purpose of this conversation: carving means turning without skidding. I don't think regular slalom courses are relevant here, as those turns are skidded. It's rare (at least where I ski and board) to see a skier carve a whole turn, and almost unheard-of to see a skier carve a whole run. With the exception of a few skis with sidecuts in the 10m range, production skis basically need to skid to be maneuverable at moderate speeds. Show me a skier (with ~20m sidecut) that can match a boarder (with sub-15m sidecut) carve for carve and I can probably show you skier who is sneaking some skids in there. The equipment differs enough that I find the claim above very hard to believe... it's like saying a good driver in a pickup truck can match a go-kart turn for turn.
There's one more factor that I haven't seen mentioned yet: softboot-wearing snowboarders generally ride with their toes and heels sticking out way past the edges of the boards. It's a side-effect of flexy boots - they need low stance angles to get leverage on the edge of the board, and overhang improves the leverage further. But as a result they can't get much edge angle without dragging their boots. It takes a tighter sidecut to get reasonable carved turns at such low edge angles. And, as a couple people have noted, most softbooting snowboarders don't carve much. Hardbooters tend to carve a lot more, and they (we) ride with little or no overhang, and our sidecuts are mostly in the 10-16m range - still significantly smaller than most skis, but at least we're comparing apples to apples now.
A couple people speculated above that a 10m radius would subject the rider to a lot of force, but that's not really true. Force is a function of speed and inclination, not sidecut radius. With a tighter sidecut, you'd lean in less to make the same turns, and go slower because you dont need to lean in as much... so you could still make tight turns with little or no extra force. It's just that for many tight turns where your old skis forced you to skid, 10m sidecut s would let you dig the edges in and carve around with no skidding.
But I think most skiers would be surprised how fun it is to truly carve every turn with NO skidding. Maybe there are people doing this now, but none of them ski where I do. With the near-20m sidecuts on most 'deep sidecut' skis, carved turns are too gradual to be very maneuverable - or you have to get really fast, inclinate a lot, and angulate hard, to get enough edge angle to make tight turns. Do-able, but it becomes a chore, and ski patrollers get on your case for going so fast.
Having gotten familiar with (and VERY happy with) a 13m radius snowboard, it really irks me that there are so few adult-length skis to choose from with sidecuts in that range. I don't care if they would look like clown shoes (great term, by the way), they would be super fun. Leaning into the turns low enough to drag your elbow requires huge speeds on 20m+ skis, but with tighter sidecuts you can do it at reasonable speeds and it's extremely fun. When you get into the 10m range such deeply inclinated turns get too tight and less fun, but I really think that tighter sidecuts, in the 13-15m range, could make it possible for more skiers to carve like the boarders in that video. Their boards have sidecuts around 13m by the way. Yes, one COULD do that on skis with radii in the high teens, but the speeds would have to be a lot higher, and the slopes would have to be a lot wider. ~13m radii would bring that kind of turning within reach of a much larger segment of the skiing population.
I got some Head Cyber iC 180s a while back with 12.6m sidecuts and I think I'll be able to drag my elbows in turns with these... I had to get them in 163cm, which is stubby for my 6 foot frame, but that was the only ski I could find with a sidecut in this range. I have only been on them one day so far and had major foot pain issues... I am looking forward to getting my boots sorted out this coming season though, I just know this setup is going to be fun.
I'm no expert in halfpipe stuff but I can air out about waist high frontside and backside, and I don't think tight sidecuts make any sense in the pipe either. The pipe just decambers the board further, tightening up the turns further. Going up the side of the pipe, I want to go straight, not turn uphill.... I really wonder if the halfpipe pros have tried boards with larger sidecuts. And if they have, I wonder what the advantage is to the small sidecuts... it makes no sense to me. Maybe they prefer to skid up the side of the pipe so the sidecut isn't a factor, I dunno. If that's the case I bet they could carry more speed between airs by carving instead... but, enough speculation. I'd love to sit down and talk with one of those folks some time.
I haven't looked at the ski market for over a year now - anyone know of some 180cm 13m radius clown shoes that I should check out?