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Path Curves (1)

Path Curves (3)

Path Curves (4)

Path Curves in the Plane (2)

As we indicated we would, we next track a point stepping on the plane as the intersection of two stepping lines,

—one red, rotating in point Y and guided by a linear measure in the bold white line, and —one white, rotating in point Z and guided by another, different linear measure in the bold red line. The bold white and red lines share point X, which is also an end-point of each of the two measures.

The stepping point traces out an invariant path curve.

Notice that it passes from Y to Z and avoids X.

If we were to take another red/white pair, we would get
another path curve on a different route between the same vertices, but with
identically the same shape.

The λ of the curve above is the ratio of the characteristic multipliers of the two measures.

A yellow line through X and the stepping point steps along with it, and cuts the bold, yellow YZ line in a range of points—which as may be seen is a third linear measure.

Now XYZ is an "invariant triangle", so the stepping point is in fact guided by three measures, one in each side of this triangle.