The intuition behind reparameterization of a curve

Question

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.

Answer 1

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.