# Approximating difference between arclength and chord length with Taylor series

Let $\alpha: I \mapsto \mathbb{R^3}$ be a regular curve. Let $P_0$ and $P$ be two vectors on this curve and $h = ||P - P_0||$ and $s$ be the arclength from $P_0$ to $P$. Prove that $|h - s| \approx s^3$.

I did this exercise previously when $\alpha$ is a circle and $h=2r\sin\left(\cfrac{s}{2r}\right)$ by expanding $|h-s|$ in it's Taylor series around $s = 0$, but I'm having trouble figuring out what to do in this more general case. I thought about expanding everything in coordinates but that would get too messy. Any help would be appreciated.

• See my answer here. – Ng Chung Tak Feb 8 '18 at 15:35
• That's another one solved, then. Thanks. – Matheus Andrade Feb 8 '18 at 17:07