Thank you Nevin for linking that article, however the author is incorrect in so many places.
Rooting the authors personal bias towards π (and I used to be one of them) by saying the maximum angle between two vectors as calculated by the dot product is 180 degrees so let’s stop at 180 degrees
and make that our fundamental value of rotation is just wrong.
The author also suggests that the cosine curve, defined from -π to π implies π as the most natural value for expressing rotations. This is incorrect
1 full rotation = 1 Tau = 1 period = 1 wavelength
Consider the wavelength example.
Waves have a period of one wavelength, not 2 wavelength halves, and so if we take the cosine curve example above it would be more intuitive if the notation embodied the idea that half the “wavelength”
of the cosine curve lies on the negative side and half the “wavelength” lies on the positive side, adding together to make one “wavelength” (period) of the curve.
One period being from 0 to τ, or from -τ/2 to τ/2 (but not - a whole π to + a whole π)
In engineering callipers measure the diameter of cylinders, and so Pi will get you to the circumference in one step, however CNC lathes don’t do this...they measure the distance of the cylinder edge from its central axis of rotation (ie measuring its radius)
so even in CNC machining, radius is now more fundamental than diameter.
I’m not suggesting we get rid of Pi, but yes, Tau is more fundamental, and if I could go back 20 years to my younger self and give him one tip it would be to use Tau.
My code is complicated enough and too often I discover a bug to be rooted in my monkey brain having put PiOverFour for 90 degrees or PiOverTwo for 180.