Asumming a curve has a natural parametrization (is simple and regular), such natural parametrization is unique or are more parametrizations equivalent to it?
I should add that by natural I mean by arc length, I was taught both terms were equivalent.
Edit:I know parametrizations in general are not unique, the question is if natural parametrizations or by arc length are an exception in that regard.
No, parametrizations are not unique.
Both $$t:[0,2\pi],z=e^{it}$$ and $$t:[-\pi,\pi],z=e^{i(t+\text{any real number})}$$ parametrize a unit circle on the complex plane.