View each track piece as a 2d vector. Add up the vectors. In a zero-tension setup, the sum is (0, 0).
As a metric for tension, assume any mismatch in position is evenly distributed. Model this as the average of all the vectors. (Thus the same displacement is more meaningful when we have fewer pieces.)
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That's the full idea. It might seem that it is ignoring rotation, because it doesn't explicitly mention rotations, but they are included because the effective vector that a track piece provides is both a current direction as well as the displacement contributed by that piece. If we wrote some code to model this, a cursor would consist of a direction (an angle) along with an (x, y) position.
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Some related math concepts:
* The exterior angles of a polygon sum to 360. So we could have another measure which is how far we are away from 360.
* Not useful in this case, but this also reminds me of winding numbers from complex analysis, which is a way to locally walk along a curve to understand which side is the "inside" or how many times a curve goes around a given point.