I can not understand how the concept of arc length works. We define
$a(t)=\int_{0}^{t}\mid\gamma^{'}(x)\mid dx$
for some curve $\gamma: I \rightarrow\mathbb{R}^2$.
If we then normalise the curve we get $a(t)=t$ which we call paramertise by arc length. I can see how the direction of the tangent is whats most relevant for the curves description, but now we change the magnitude of the tangent.
What does this mean and how does it matter? Intuitively Id say that this has to do with "speed". But how does one make sense of this? Is it the lenght of the intervall $I$ ? because any intervall has the same number of points.