Guys, those two montages of Svindal and Vonn do show quite a contrast between extension and retraction through the transition. The reasons really don't matter--whether there was a bump in Lindsey's transition, for example. What matters is that the circumstances dictated the movements they used.
And I suggest that that is the critical point. They did not "choose" to do "an extension turn" or "a retraction turn" (somewhat more descriptive of the transitions than "cross-under/over," at the very least). Nor do I, when I'm skiing (other than when intentionally demonstrating one end of the spectrum or the other). The truth is that good skiers flex or extend their legs and body as needed, continuously, throughout all phases of the turn, primarily to manage the forces that influence the motion of their center of mass and the pressure on their skis.
In a simple example, when faced with a large bump or a freestyle "kicker," we flex or extend, subtly, vigorously, or not all, according to the outcome we desire. The two "ends of the spectrum" would be completely absorbing it so that it does not throw us upward at all, vs. vigorously extending and leaping to get as much air as possible. In between would be the entire range of subtleties that allow us to choose just how much "air" we get, and how far we'll fly. When you're running through the woods and come to a fallen tree across your path, or a small creek you need to hop across, you employ flexion and extension movements as appropriate to get across the obstacle. You would not "choose" "either" to flex or to extend in advance--it's an on-the-spot live, unconscious adjustment based on skill and experience, timed, directed, and metered out according to the unique needs of the moment. It might involve pulling your feet up (retracting) sufficiently to clear the tree like a hurdler, or jumping a little higher and then pulling your feet up, or leaping just far enough to clear the creek. It certainly isn't just "one or the other." Really, it's just all part of running through the woods, and it's the genius of our bodies that makes the "decisions" and adjustments.
And so it is with these retraction and extension movements. Identifying the extremes of the spectrum and practicing them as drills certainly helps develop skill and range. But it would be folly to mistake those extremes as a simple two-position "switch" that you need to choose consciously when skiing.
As a final example, visualize the analogy of how our skis literally throw our bodies back and forth between turns like two people tossing a medicine ball. Imagine doing that--you and a friend stand some distance apart, and throw the ball back and forth. So...how high do you throw it? Depends on a lot of things, doesn't it? How far apart are you? Is your partner standing above you or down in a hole? How fast will you throw it--a flat, laser-like bomb, or a high, slow lob, or somewhere in between?
As the following video and animation attempts to portray, this game of toss is a very good analogy for the transitions of ski turns, where our skis, combined with precisely controlled and directed flexion and extension movements, "throw" our body (center of mass) from one turn to the next. Sometimes it's a flat trajectory ("cross-under," if you must), sometimes it's a high, slow lob (call it "cross-over," if that works for you). Always, it's somewhere on the spectrum, dictated not by preference or conscious choice, but by the specific needs of the moment.The ends of a spectrum define the range. Cutting out the middle section may bring some sort of simplicity. Kind of like cutting out your heart to "simplify" your anatomy....
PS--aside to JASP--the presence of a photographer does not define a frame of reference. Indeed, a camera is a great way to visualize more than one frame of reference. Give each of CGeib's ants a camera, and you'd have two very different views of what's going on. Download a chairlift and video a skier traveling directly below you, and the resulting image would show the skier "not moving" at all--simply staying in one place in the middle of the screen as the snow "moves" across the screen. Conversely, put the camera on a tripod and shoot a downhill racer racing past, and all you'll see is a blur against the "stationary" background of the slope and the mountains. But even when the videographer is standing still on the same slope as the skier, it does not mean that the image will represent the so-called "inertial frame of reference" of the videographer. Indeed, good videography usually dictates that you hold the camera steady and "still" on the skier, minimizing as much as possible his movement on the screen.
And even if you just mount your camera on a tripod to represent the "inertial" frame of reference of a stationary observer while the skier skis through the frame, it would not clarify the "difference" between "cross-over" and "cross-under," as I believe you have suggested. It would instead show the two paths of the body and and the skis simply crossing each other, neither one being "still" as the other moves above or beneath it. That's what we see in the two montages of Vonn and Svindal. The more definitive difference between those sequences is, as has been much described, the flexion and extension movements of the two skiers in these particular turns.
"Cross-over"? "Cross-under"? Or just a turn transition?
Clearly neither one or the other...just a transition...I do think the original question jamt asked (what's the difference between over and under?), has generated some useful conversation, and that we have all discovered that speaking in extremes regarding technique is actually a good thing (round-about though it may be) and will eventually lead us, the skier, to the middle ground (eg. these transitions) as well