ArcRefinement
As n increases, the purple waveform refines its arcs to approach the straight line segment NP.
All waveforms here are composed of half-circle arcs of diameter d, whose each arc length is
.
The basic waveform #0 has two arcs, the negative blue & the positive red, spanning the wavelength |NP| = 4 = 2d0, where the diameter of each arc d0 = 2, hence its total arc length is
.
Then each arc of #0 is divided into n pairs of negative-positive arcs, yielding the purple refined waveform #n containing totally 4n arcs, spanning the total length |NP| = 4 = 4n⋅dn. That means each arc's diameter dn = 1/n, and the total arc length of #n is
.
That means the total arc length of the refined waveform #n is always equal to the arc length of the basic waveform #0, which is 2π, even when #nn→∞ approaches the straight line segment NP:
.