3
$\begingroup$

There is this supposed to be not so hard question but concept of re-parametrisation by arc length bothers me a bit.

Let $\alpha:I\to R^2$ be a regular paramatrised plane curve (arbitrary parameter) and let $n(t)$ be its normal vector.

I was trying to show that $$\langle\alpha'(t)-\frac{k'(t)}{k(t)^2}n(t)+\frac{1}{k(t)}n'(t),\alpha'(t)\rangle=0$$

My approach:

$\langle\alpha'(t)-\frac{k'(t)}{k(t)^2}n(t)+\frac{1}{k(t)}n'(t),\alpha'(t)\rangle=\alpha'(t)\cdot\alpha'(t)-\frac{k'(t)}{k(t)^2}n(t)\cdot\alpha'(t)+\frac{1}{k(t)}n'(t)\cdot\alpha'(t)$

$=|\alpha'(t)|^2-\frac{1}{k(t)}k(t)n(t)\cdot n(t)$

$=|\alpha'(t)|^2-1$

And it will equal to 0 if the curve is parametrised by arc length.

My questions are:
1. Do we have to reparametrise the curve by arc length in the first place? If so, how should we do that?
2. If we did not need to reparametrise the curve by arc length, how can we say that $|\alpha'(t)|^2-1=0$?

I think this problem is almost done but just that one last step. Can anyone please clarify it?

Helps are greatly appreciated! Thanks!

EDIT

I just realised it is related to this question

Tangent of evolute and singed curvature

But I am still unable to show the last step! Thanks.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.