Curvature of the moment curve

Question

I would like to compute the curvature of the moment curve in $\mathbb R^n$: $$ (s,s^2,s^3,\dots,s^n). $$ I am especially interested in the curvature at $0$. But this is not parametrized by arc length. I also know a formula $$ k(s)=\frac {|\gamma'(s)\times\gamma''(s)|}{|\gamma'(s)|^3}. $$ which works in the case $n=3$ only. I wonder if there is also a higher dimensional version of the above.

Answer 1

Let's say your curve is $\gamma(s): [a,b] \to \mathbb R^n$. Then we can find the the legnth of the curve on the interval $[a,t]$ using the formula

$$l(t) = \int_a^t \Vert \gamma'(s)\Vert ds$$

Note that if $\gamma$ is sufficiently well behaved then $l$ is strictly increasing and therefore has an inverse. Now define

$$\tilde \gamma (s) = \gamma(l^{-1}(s)).$$

It isn't difficult to see that the reparametrization $\tilde \gamma$ of $\gamma$ is now parametrized by arc length, so the standard formula for the curvature can be used.

Alternatively there is a more explicit formula

$$\kappa = \frac{\sqrt{ \|\gamma'\|^2 \|\gamma''\|^2- \langle \gamma', \gamma''\rangle^2 } }{\|\gamma'\|^3}.$$