So I have this curve $$ \alpha(t) = (t−sin(t), 1−cos(t), 4 sin( \frac{t}{2}))$$ which is not parametrized by arclength. And I need to find if the curve $$\gamma(t) = \alpha(t) + k_{\alpha}(t) N_{\alpha}(t) $$ is contained on a plane. I've been trying to solve this problem by finding the curvature and normal vector of $\alpha$. However it's getting too complicated and I don't know if there is an easier and better way to approach this.
Thanks in advance.