Performance skiing physics 101
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Here's one way to look at and help you understand the physics of performance skiing. One of the harder things to understand seems to be how the reaction force creates circular motion and how that relates to performance skiing.
Since skiing is all about turning and circular motion, let's dive in.
Some terminology
Deflection
We refer to deflection generally as being deflected from a straight path, onto a circular path. Deflection is what moves skiers across the hill.
You may hear this mentioned as "deflection of COM" - to emphasize that we're talking about the skier's mass moving across the hill, rather than just say, his or her boots!
Offset
Offset is related to deflection and basically refers to how far apart the turns are, laterally. In racing, this is given by the relative position of subsequent gates. In free-skiing, this is chosen by the skier.
Wiggle in the fall line refers to skiing without much offset, without much deflection of the COM across the hill. This is not performance skiing!
Reaction force and circular motion
Run straight at a wall: the sudden reaction force from the wall will stop you.
Run at a wall at an angle: the reaction force from the wall will deflect you.
Tie yourself to a stake and run straight out. The sudden reaction force from the rope will stop you.
Tie yourself to a stake and run, but not away from the post, but in a different direction, so that the reaction force won't be straight against the direction of movement: the reaction force (when the rope ends) will deflect you, with a quick circular acceleration.
Continue to run while the rope is tensioned. If you run straight outwards from the post, you'll just wear out your shoes and not move at all. If you run at a bit of an angle to the rope, you'll end up deflected in a circle.
Gravity is like you running in the same direction. You can't change the direction of gravity but you can change the wall's or rope's direction and reaction. We do that by putting the skis on edge and let them interact with the snow.
Ski down the slope. Throw the skis sideways and the reaction force from the snow will stop you (hockey stop).
Ski down the slope. Put the skis on edge at an angle, so not quite sideways: the reaction force from the snow will deflect you gently into a turn. As gravity keeps pulling you down, you will keep turning until you loose momentum. If you had instead kept running or had rockets strapped to your boots, you'd keep turning up the hill and in a circle.
The more you put the skis on edge, the reaction force will be stronger, redirecting you faster into a tighter turn.
As you start carving, depending on the hardness of the snow, you can put as much as 2-3g of force into it. The stronger reaction force will interact with the sidecut design and bend the skis into a tighter turn.
Pivoting vs carving and deflection
Pivoting and steering are like running into a wall of gell-o or tied with a soft rubber band: it will slow you down some, but not quite deflect you. To be deflected across the hill, you need a strong platform, i.e. carving. By pivoting, you'll slow down slowly and change direction of COM slowly. By putting the skis at a sufficient angle to carve, the rate of slowing down (in the "down" direction) is faster and the change in direction (deflecting) is much faster.
Technique
Since the name of the game is "how fast we deflect", the speed at which we establish the serious edge angles matters. I.e. the faster you get on a solid edge, the faster you turn... just ask Ted. That is technique.
The degree of inclination vs angulation is a matter of two things: putting the skis on edge at a sufficient angle for the deflection you want to get and balancing against it while you turn. It also has the benefit of increasing edge hold.
I'm sure I missed like a thing or two. Or more.
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