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The problem with pi vs tau is that it is pretty much impossible to convince everybody that one of them is the best and also that there are no other better choices. So, in the absence of unanimity, the incumbent pi wins by default. As an example of the diversity of opinions in this matter, while I agree that tau is better than pi, I believe that an even better choice than either of them is pi/2. Any choice of the constant is equivalent to a choice of the measurement unit for the plane angle. Choosing tau is equivalent to choosing the cycle as the measurement unit for the plane angle ("cycle" was actually a standard name for this unit of plane angle, even if it is now out of fashion, e.g. in the old literature there are many references to cycles per second, cycles per meter etc.), while choosing (pi/2) as the constant, is equivalent to choosing the right angle as the measurement unit for the plane angle. There are several arguments why the right angle is a more appropriate unit than the cycle, besides the fact that the right angle was already the unit of plane angle used by Euclid. Even in a plane, but much more so in 3D space, a rotation of 1 cycle is ambiguous, it is not associated with a certain axis of rotation, like a rotation of 1 right angle. Given a rotation of 1 right angle, a rotation of an arbitrary angle is just the rotation corresponding to a multiple of that right angle, which can be non-ambiguously defined, like you expect for any physical quantity, to have a value that is a multiple of its unit. Also in the trigonometric functions, the right angle is the most convenient unit for the arguments (making their reductions equivalent with taking the fractional part) and I am dismayed that the floating-point standard has chosen to recommend the functions sinPi, cosPi and the like, to be included in the standard libraries of the programming languages, instead of the much more convenient functions where the angles are measured in right angles, i.e. sin((pi/2)*x) and so on (such functions do not have the argument reduction problems that plague the functions with arguments in radians; the former are more convenient for large angles, while the latter are more convenient for very small angles). |
Could you explain why a cycle is ambiguous in 3D? It seems like for any choice of coordinates, one cycle would always entail a return to an original point. But I'm probably misunderstanding.