|...oversimplified "average angle" that I tried to get away with...
Nope--you can't get away with ANYTHING around here!
Nice explanations, Tom. To me, the only thing that makes sense, at least "practical" sense, is to consider only what happens directly beneath my foot, where the forces that apply to me actually apply--the point of the ski over which I'm balanced, and through which the "line of action" applies--where "action meets reaction." The line of action indicates the direction of the forces that apply to me, and in a (theoretical) circular pure-carved arc, that direction is directly sideways, perpendicular to the direction I'm traveling at any given moment. And the ski, of course, is going the EXACT direction I'm traveling--at this point beneath my foot--representing a ZERO steering angle.
The rest of the ski--tip and tail--really doesn't affect ME. It can influence that steering angle under my foot in several ways, but it does not affect ME. It really doesn't matter what happens to the rest of the ski as far as its effects on me go--it can tighten or straighten its arc, as long it doesn't change that angle beneath my foot--witness the incredible flapping and gyrations of the ski in a slow motion film of a downhill racer. It doesn't matter how LONG the ski is, either, although the angle of the tip of a longer ski in the same carved arc is obviously different than the angle of a short ski. Again, the only steering angle that matters is the angle at the point I'm standing on, the direction the ski pushes on ME.
Of course, in that theoretical perfect carve again, the steering angle changes constantly, as does my direction of travel, as I go through the turn. I like the image of a ball-bearing rolling around in a bowl. The circular sides of the bowl provide a continuous centripetal force, pushing the ball at right angles to its direction of travel (at any instant). The tangent of the bowl's arc at the point where the ball contacts it represents the direction the ball is traveling at that moment, and (obviously) also represents the steering angle. The bowl is like a very long ski, so long that it bends 360 degrees and describes the entire circle.
Another applicable image is the ball on a string, swinging in a circle. Here, the STRING represents the centripetal force. Obviously, it always pulls directly toward the center of the circle, along the line of the string (the only direction it CAN pull). And that direction is always perpendicular to the direction the ball is traveling, represented by the tangent of the arc--which again represents a ZERO steering angle.
Perhaps these illustrations can help:
Good theoretical stuff. I'm not sure it lends any more real insight into how to start a turn than the little side-track about teaching that we switched to above, but it's good to understand!