Here is a question i am not sure how to tackle, I am not familiar with how to deal with family of curves and don't really have much time to look around for the definition as i am sitting the exam in two days.

The question is divided into three parts: Here is my attempt any help appreciated.

1) I am have no idea, i think it is a case of knowing the definition and i don't.

2) It is simply constraining the local parametrization to the given function so:

$\displaystyle xz-hy=> v*sin(u)=h(1-cos(u))=> h= v*sin(u)/(1-cos(u))$ which is a constant.

3) $\displaystyle \psi(u,v)=const$ is like phi therefore the tangent vectors to the family defined by the psi are of the multiples of $\displaystyle \psi_{v}x_{u}-\psi_{u}x_{v}$

So for the families to be orthogonal their tangent must be orthogonal and so$\displaystyle (\psi_{v}x_{u}-\psi_{u}x_{v}).(\phi_{v}x_{u}-\phi_{u}x_{v})=0$

Using the fundamental forms E=1=G and F=0 we get $\displaystyle \psi_{v}\phi_{v}+\psi_{u}\phi_{u}=0$ which after differentiating gives $\displaystyle \psi_{v}\sin(u)-\psi_{u}v=0$

And after that i am stuck ... any help would be appreciated.