> In my view, because pi crops up unavoidably in math, if you concoct a "unit" to get rid of pi in one place, it will simply crop up somewhere else, perhaps in a denominator.
That doesn't mean you shouldn't try to put it in a convenient place.
One way to think of the post is: where you want pi to come up?
With arc length parametrization f(r) = (cos(r), sin(r)), it comes up in the parameter space (one turn: 0 ). If you had the whole thing in terms of turns, you'd instead have (as a primitive) some kind of function g(t); with one full round for 0 . It'd then have to be true that
f(2 pi t) = g(t) = (cos(2 pi t), sin(2 pi t)).
Pi would come up in the velocity:
f'(r) = (-sin(r), cos(r)) = if
(i u means rotate the vector u by 90 degrees counter-clockwise)
g'(t) = 2 pi f'(2 pi t) = 2 pi (i f(2 pi t)) = 2 pi (i g(t))
Before, you had |f'| = 1. Now you have |g'| = 2 pi.
For classical physics (kinematics and dynamics) applications and classical geometrical applications (curvature, etc), it's really convenient to have that speed term (|f'|) being 1. This is one of the major motivations for arc length parametrization.
By the way, this can't be understated. It really simplifies kinematics, dynamics, geometry, etc, having |f'| = 1 throughout. It's not just for circles. This can be done for an extremely large class of curves and it makes the related math much more understandable and easier to deal with.
For a lot of computer graphics (I believe this is where Casey comes from), you care less about tradicional mathematics for physics and geometry. So you'd rather (maybe) take this pi appearing in the parameter space and push it to the velocity.