whynothis
Sep8-08, 04:41 PM
1. The problem statement, all variables and given/known data
Given a parametric curve, \alpha(t) = (x(t),y(t) ), not necessarily arc length parameterized show that the curvature is given by:
k = \frac{x'y'' - y'x''}{|\alpha'|^{3}}
2. Relevant equations
As I understand this the curvature is defined from the point of view of a arc length parameterization as
k = |\frac{d^{2}\alpha(s)}{ds^{2}}|
So I tried using the chain rule...
3. The attempt at a solution
\frac{d\alpha}{ds} = \frac{d\alpha}{dt}\frac{dt}{ds} where dt/ds = 1/(ds/dt) = 1/|\frac{d\alpha}{dt}(t)|
Then I differentiated again to get...
\frac{d}{ds} \frac{d\alpha}{dt}\frac{1}{|\frac{d\alpha}{dt}(t)| } = \frac{d}{dt}\frac{dt}{ds} \frac{d\alpha}{dt}\frac{1}{|\frac{d\alpha}{dt}(t)| }=\frac{d^{2}\alpha}{dt^{2}}\frac{1}{|\frac{d\alph a}{dt}(t)|^{2}}}-\frac{d\alpha}{dt}\frac{\alpha'\cdot\alpha''}{|\fr ac{d\alpha}{dt}(t)|^{4}}
This isn't what I should get. It seems that I am clearly missing something since all the problems I try and do along these lines I am stuck on. Does anyone have any insight on this?
Thanks in advanced
Given a parametric curve, \alpha(t) = (x(t),y(t) ), not necessarily arc length parameterized show that the curvature is given by:
k = \frac{x'y'' - y'x''}{|\alpha'|^{3}}
2. Relevant equations
As I understand this the curvature is defined from the point of view of a arc length parameterization as
k = |\frac{d^{2}\alpha(s)}{ds^{2}}|
So I tried using the chain rule...
3. The attempt at a solution
\frac{d\alpha}{ds} = \frac{d\alpha}{dt}\frac{dt}{ds} where dt/ds = 1/(ds/dt) = 1/|\frac{d\alpha}{dt}(t)|
Then I differentiated again to get...
\frac{d}{ds} \frac{d\alpha}{dt}\frac{1}{|\frac{d\alpha}{dt}(t)| } = \frac{d}{dt}\frac{dt}{ds} \frac{d\alpha}{dt}\frac{1}{|\frac{d\alpha}{dt}(t)| }=\frac{d^{2}\alpha}{dt^{2}}\frac{1}{|\frac{d\alph a}{dt}(t)|^{2}}}-\frac{d\alpha}{dt}\frac{\alpha'\cdot\alpha''}{|\fr ac{d\alpha}{dt}(t)|^{4}}
This isn't what I should get. It seems that I am clearly missing something since all the problems I try and do along these lines I am stuck on. Does anyone have any insight on this?
Thanks in advanced