Say $a(s)$ is a regular curve with curvature $k$. Then it's local projection onto its osculating plane at $P$ is given by $\gamma(s)=(s,\frac{k_os^2}{2})$, where $k_0$ is the curvature at $P$
We can calculate the curvature of $\gamma(s)$ by the equation:
$\frac{|\gamma'(s) \times \gamma''(s)|}{|\gamma'(s)|^3}$ and according to the statement of the problem, this value should equal $k_o$. Well when I do this calculation I get that the denominator does indeed equal $k_o$, but in the denominator I get $1+3k_os+3(k_os)^2+(k_os)^3$ when I expand everything out.
Dang it.
Is there some reason that the denominator should be equal to 1 that I'm not thinking of? I mean, it can't be true that the projection is arc length parameterized if the original curve is, right?
Am I approaching this problem wrong? any feedback appreciated.