There are poorly-made turns happening out there. But they are still "turns" with "transitions." The line skiers make moves them left-right across the hill. I'd go with this as the definition of turn, as
@François Pugh has already suggested.
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My definitions are:
"turn" - repeating left-right travel down the hill
"transition" - skier actions between fall lines that cause the switch from going left to going right
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My concern is that this thread will ignore the reality of poorly-made turns and come up with a bunch of ideal turn descriptions and definitions. One side-effect of that will be people saying the skis have to get onto new edges to start the new turn, and the CoM has to cross downhill over the skis to the inside of the new turn to start the new turn.
In most poorly-made turns that I watch from the chair, I see skiers avoiding moving the CoM downhill over the skis by any means possible. These skiers avoid spending any time on downhill edges above the fall line, and in most turns these skiers don't get their skis flat until they point downhill. The left-right travel change gets going far before the skis point downhill, so these turns start on old edges and stay on "flattening" edges through the whole top-of-turn. New edges happen at the bottom half of the turn, which at that point are uphill edges.
On steeper groomers with hard sonw, these skiers stay in the back seat prettymuch the whole time they ski. They lighten and pivot their skis to point downhill or all the way around in the new direction without ever moving their CoM over the skis. They get onto new edges after the fall line, which mean those new edges are uphill edges. They do not know what it feels like to travel along on downhill edges. But, they are making turns. They are moving them along a z-shaped line down the hill.
On shallow groomers, these skiers simply avoid completing their turns. They travel down the hill in a narrow corridor, leaning their bodies left-right to make slight turns. Their CoM does indeed get "inside" the new turn, but that doesn't mean they move downhill over their skis, and it doesn't mean their skis tip to flat then to downhill/new edges above the fall line. They move from left edges to right edges. These skiers can ski a narrow corridor this way, from the back seat, on shallow pitch terrain because the speed never gets too high for them. Such "needs improvement" turns could never be made on steeper hard-snow groomers by these skiers. I affirm that these are still turns, just not ideal.
One of the challenges of teaching intermediates on hard NE snow is to help such resistant skiers lose their fear of getting onto their new downhill edges, with CoM on the other side of the skis, before the fall line.
There are also
good turns that stay on old, uphill edges through part of the top of the turn. Think about it. In pivot slips, the skis don't get flat until they both point downhill. New edges happen after that when they turn to point the new direction. The skis go from downhill edges to downhill edges in a pivot slip, and the amount of that edging controls the speed of the downhill travel. The turning of the skis is done by skier muscle action, not ski-snow interaction. Skidded turns are a modification of this mechanism, and as such involve getting the skis to point downhill using muscle action ("femur rotation") while the skis are flattening. Extension of the new outside leg contributes to the skis turning to point downhill on flattening old uphill edges. These skidded turns are not necessarily seen as "needs improvement" turns, unless you are a devotee of PMTS.
Any general definition or description of "turn" and "transition" needs to take these skidded turns into account. It also needs to take into account bump turns, and even competitive mogul skier turns. So the fewer words in these descriptions/definitions, the more inclusive they will be.
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To discuss specifics, I'd suggest we specify which turns in which conditions and on what terrain we are talking about. Suggestion:
--We clarify "needs improvement" transition mechanics from "excellent" transition mechanics so our definitions don't mean all these recreational skiers are not actually making turns at all.
--We specify what type of terrain we are talking about - seeded mogul run, trees with polished bumps, tight trees with fresh 12" powder, winched groomer with hard snow, etc.
--We clarify whether we are talking about skidded turns, carved turns, narrow corridor turns, wide corridor turns, etc.