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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 ;)

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1 Answer 1

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All that's going on here is that $\alpha'(t)$ and $n(t)$ give an orthonormal basis for $\Bbb R^2$, so $$\alpha(t) = \langle \alpha(t),\alpha'(t)\rangle \alpha'(t) + \langle \alpha(t),n(t)\rangle n(t).$$

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  • $\begingroup$ could you please explain me how did you get the identidy above ? $\endgroup$
    – Ava3141
    Commented Aug 21, 2016 at 23:09
  • $\begingroup$ Have you studied any linear algebra? If $e_1,\dots,e_n$ is an orthonormal basis for $V$, for any $v\in V$, write $v=\sum\limits_{j=1}^n c_je_j$ and dot both sides with $e_i$. $\endgroup$
    – Ted Shifrin
    Commented Aug 21, 2016 at 23:14
  • $\begingroup$ I took a linear algebra course 3 years ago. Thanks so much ! $\endgroup$
    – Ava3141
    Commented Aug 21, 2016 at 23:40

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