The circle is a shape, distinct from a square or a triangle or a straight line. I see people making triangular or Z-shaped turns, square turns (hockey slides), and straight lining. Can you see in your mind's eye how a circle is the best shape for the turn, compared to the others?
(When I talk about the shape, I am talking about the directional line the skis make in the snow...)
Visualize having four long sections of rope and a slope. Lay one rope down to demarcate a straight path, aka the fall line. Then lay a rope down in a Z-path and another in a right angled path. Can you see where the skier needs to leave one straight line abruptly to follow the other straight line?
Now take the last rope and lay it next to the others as a series of S-es down the slope. See how the curved arcs of the rope allow the skier to stay on one continuous line by smoothly and gradually changing edges to change the direction of the curve left and right?
Another idea is that when the skis are going straight down the fall line, they are directly in the path of gravity. When they are across the fall line, they are completely out of the path of gravity and therefore stopped. In a round turn, a skier spends about the same amount of time in and out of the path of gravity.
Draw a box in the snow, and round off the corners to make a circle. Moving gradually in and out of gravity's path is the secret to the circle. It's the secret to smoothness, by evening out the starts and stops that are inherent in each turn.
This is what I call guidance and steering, going from one set of edges to the other and keeping the edges engaged as long as it takes to keep a consistent rate of descent and maintain rhythm, and no more.
Rounding out railroad tracks is how I like to develop a sense of the skis' preference for making circles over any of the other shapes.
The curves are arcs of a series of imaginary circles going down the slope. Keeping consistent circles is a PSIA-type task, as is doing a series of one size circles and changing in the space of one turn to a set of different size circles, often corresponding with a terrain change that makes the change of tempo appropriate. Often the additional request to not change speed is given.
I also said that making circles is a technical thing, but the size of the circle is tactical. The PSIA task I described involves both physical technique and tactical thinking.
I'm not PMTS-trained so I'll let that gang speak for themselves.