Questions tagged [arc-length]

Given $t\in I$, the arc length of a regular parametrized curve $\alpha : I \rightarrow {\mathbb R}^3$, from the point $t_0$, is by definition $$ s(t)=\int_{t_0}^t |\alpha'(t)| dt, $$ where $$|\alpha'(t)| =\sqrt{ (x'(t))^2+ (y'(t))^2+(z'(t))^2} $$The generalization to $\mathbb{R}^n$ is immediate. In particular, if $n=2$ and $\alpha$ lies on some function $y=f(x)$ with $\alpha(t_0)=(a,f(a))$ and $\alpha(t)=(b,f(b))$, the arc length along $f$ from $a$ to $b$ is $$ \int _a^b \sqrt{1+(f'(x))^2} dx $$ Length of curve is independent of parametrization, so for a calculation related with a curve, for instance, curvature, torsion and so on, we want to find a suitable parametrization. If $|\alpha'(t)|=1,$ then $\alpha$ is a curve parametrized by arc length $t$.