Can anyone help me with finding a parametrization of the pedal curve of a regular arc lenght parametrized curve $\alpha(t)$? I looked at some books and I found 2 parametrizations of it:
(1) $\beta(t)= \alpha(t) - \langle\alpha(t),\alpha'(t)\rangle\alpha'(t)$
(2) $\beta(t)=\langle\alpha(t),n(t)\rangle n(t)$
I understood the first one, but I didn't understand how to get the second.
Also, how can I show that if $\alpha(t) $ does not pass through the origin $\beta(t) $ has singularities right on the inflection points of $ \alpha$?
I had tried to solve this question by taking the derivative of (1), but the hypothesis that $\alpha $ doesn't pass by the origin seems to be useless.
Thanks in advance ;)
All that's going on here is that $\alpha'(t)$ and $n(t)$ give an orthonormal basis for $\Bbb R^2$, so $$\alpha(t) = \langle \alpha(t),\alpha'(t)\rangle \alpha'(t) + \langle \alpha(t),n(t)\rangle n(t).$$