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How can I able to parametrize the curve $x^2=4ay$ such that it becomes a
($i$) it becomes a regular curve.
($ii$)the parametrization becomes a unit speed parametrization.

Actually I want to find the signed curvature of the parabola given curve.To do so I need to parametrize it and exactly where I have stuck.

Can I get some help?

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  • $\begingroup$ what do you mean 'regular'? what is exotic about the curve guven? $\endgroup$
    – Guy
    Mar 11, 2014 at 11:39
  • $\begingroup$ If $\gamma(t)$ is a parametrized curve then it is call regular if $\gamma'(t) \ne 0$ $\endgroup$
    – jigja
    Mar 11, 2014 at 11:42
  • $\begingroup$ Parametrize by $x=2at,y=at^2$ (and assume $a\neq 0$). $\endgroup$ Mar 11, 2014 at 12:01

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Georges' parametrization is fine, but you won't find a unit speed parametrization in simple terms. Fortunately you don't need it, but you want to find the curvature $\kappa$ of the curve $c(t):=(2at,at^2)$ instead. Now that's given by $$\kappa=\frac{\det(c',c'')}{\|c'\|^3}.$$ You might find How to reparametrize curves in terms of arc length when arc-length evaluation cannot be computed analytically helpful as well.

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