Curvature of a curve whose unit tangent creates a constant angle with z-axis I have this question:
Consider a curve $\gamma$(s) and its projection to the plane $\beta$(s), i.e
$$\gamma(t) = \begin{bmatrix} f(s)\\ g(s) \\ h(s) \end{bmatrix},  \beta(t) = \begin{bmatrix} f(s) \\ g(s) \\ 0 \end{bmatrix} $$
Where s is the arc length parameter for $\gamma$.


*

*Is s also the arc length parameter for $\beta$?

*Let $T$ be the unit tangent vector of $\gamma$. Prove that the angle between $T(s)$ and the z-axis is at least $\frac{\pi}{4}$ iff $|h'(s)|\leq \frac{1}{\sqrt 2}$.

*Assume $|h'(s)|\leq \frac{1}{\sqrt 2}$ holds for all $s$. Assume there is a constant C so that $$\kappa_{\gamma}\leq C$$ is satisfied everywhere. Prove that the curvature $$\kappa_{\beta}\leq 100C.$$
It would be great if someone could help me with this problem because I have been working on it for quite some time and have no idea how to arrive at the conclusion for part 3. Thank you in advance!

An alternative answer: $$k\gamma=\frac{\mid g'h''-h'g''h'f''-f'h''f'g''-g'f''\mid}{((f')^{2}+(g')^{2}+(h')^{2})^{3/2}}$$ greater and equal to C #.
Thanks to Hieu Hoang who wrote this short equation.  
