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I have long had this unsolved question about arclength parameterization in my head and I just can't bend my head around it. I seem not to be able to understand why velocity with arclength as the parameter is automatically a unit tangent vector. My professor proved in class that

s(s) = s = ∫ ||v(s)|| ds with lower and upper bound s_{o}and s_{1}

He said that by fundamental theorem of calculus, ds/ds = 1 = ||v(s)||, so the velocity vector with respect to arclength s is always a unit tangent vector.

I understand the mathematical proof the professor did but I really need an intuitive explanation. I have thought about this situation where velocity is represented by distance in one dimension, but it seems that it is not always the case that V(s) will be unit speed. Then why is it the case when velocity is parameterized in terms of arclength so that the speed automatically becomes 1?

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# I Velocity with respect to arclength is a unit tangent vector?

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