What is the intuition behind reparameterization of a curve? I understand it as layers of parametrization, when we add the composition operation, we create a chain of parameterization as long as we want, this chain we can see it as ”factors” that controls the speed of the main parameterization.
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1You trace out the same underlying curve, moving in the same direction at each point, but you change the "speed". – Ian Sep 14 '20 at 19:06
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The point does not change but speed, acceleration with time dependence can arbitrarily change. Like the same speedway for all competing supercars including a repair road-roller.
But differential geometric slope, derivatives etc. that isometrically depend on the first fundamental form are invariant.
Narasimham
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