arclength parametrization intuition

Question

I have a question about parametric curves. I have learnt about arc length re parametrization and I understand how do the problems , for example finding the length of the vector and integrating with respect to another variable and solving for it. However I feel like I don't fully understand the geometric intuition of this , like what it really is doing and what It means ? Could anyone help shed some light on this ?

Why is it that it is common to define curvature with respect to arc length for example?

Thank you in advance.

Answer 1

This may be helpful to you: How (and why) would I reparameterize a curve in terms of arclength?

And here's a derivation of the fact that the velocity under arclength parametrization is a unit vector. This property shows the behavior of this parametrization to admit intrinsic properties of curves. From here we may build formulas for curvature and torsion - two characteristics that uniquely determine each curve.

\begin{align} &s = \int_{0}^{t} \mid\alpha^{'}(t) \mid \\\\ &\frac{ds}{dt} = \mid\alpha^{'}(t) \mid \\\\ &\frac{dt}{ds} = \frac{1}{\mid\alpha^{'}(t) \mid}\\\\ &\alpha^{'}(s) = \frac{d\alpha}{ds} = \frac{d\alpha}{dt}\frac{dt}{ds}= \alpha^{'}(t)\frac{1}{\mid\alpha^{'}(t) \mid}\\\\ &\mid \alpha^{'}(s) \mid = \mid \alpha^{'}(t){\frac{1}{\mid \alpha^{'}(t)\mid }\mid} = 1. \end{align}