Curvature of a curve $\vec{r}(s)=(3+s)\hat{x}+\hat{y}$

Asked Oct 06 '17 at 23:22

Active Oct 06 '17 at 23:22

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I am having problem in calculating the curvature of the following curve.$$ \vec{r}(s)=(3+s)\hat{x}+\hat{y} $$ where $s$ is the arc length parameter.

I know that $$ \kappa=\left|\frac{d\vec{T}}{ds}\right|=\left|\frac{\vec{T}'(t)}{\vec{r}'(t)}\right|=\frac{| \vec{r}'(t)\times \vec{r}''(t) |}{|\vec{r}'(t)|^3} $$ But how to calculate in terms of $s$?

asked Oct 06 '17 at 23:22

I am pi

Same formula. You'd better hope that $|\vec r'(s)|=1$ if $s$ is really an arclength parameter. – Ted Shifrin Oct 06 '17 at 23:30
So it will be $\kappa=\frac{| \vec{r}'(s)\times \vec{r}''(s) |}{|\vec{r}'(s)|^3}$ – I am pi Oct 06 '17 at 23:38
Sure. The first formula works for any parametrization (i.e., all parametrizations). – Ted Shifrin Oct 06 '17 at 23:50
Then the curvature will be zero as $\vec{r}''(s)=\vec{0}$ – I am pi Oct 07 '17 at 00:21
Yes, I am pi. Indeed, this is a parametrization of a horizontal line in the $xy$-plane, and lines don't curve!! :) – Ted Shifrin Oct 07 '17 at 00:24

Curvature of a curve $\vec{r}(s)=(3+s)\hat{x}+\hat{y}$

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