Curvature

**mathman88** · September 12th 2009, 11:25 AM

Compute explicitly a plane curve $c(s)$ which is parameterized by arc length and whose curvature is $\kappa(s) = s^{-\frac{1}{2}}$ .

**NonCommAlg** · September 13th 2009, 01:33 AM

Originally Posted by mathman88

Compute explicitly a plane curve $c(s)$ which is parameterized by arc length and whose curvature is $\kappa(s) = s^{-\frac{1}{2}}$ .

we have $c(s)=(x(s), y(s)),$ where $x'(s)=\cos \theta, \ y'(s)=\sin \theta,$ where $\frac{d \theta}{d s} = \kappa = \frac{1}{\sqrt{s}}.$ thus one solution would be $\theta = 2\sqrt{s}$ and so $x'(s)=\cos(2\sqrt{s}), \ y'(s)=\sin(2\sqrt{s}).$

now integrate (substitute $2\sqrt{s}=t$ and then use by parts) to find $x(s)$ and $y(s).$

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