Family of surfaces (Diff Geometry)

[Tchakra]

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.

Homework Statement

(this link has an image of the problem)
img126.imageshack.us/img126/807/diffgeombs3.png

Homework Equations

hmmm, anything from differential geometry ie fundamental forms, theory of surfaces ...

The Attempt at a Solution

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

3) $\psi(u,v)=const$ is like phi therefore the tangent vectors to the family defined by the psi are of the multiples of $\psi_{v}x_{u}-\psi_{u}x_{v}$
So for the families to be orthogonal their tangent must be orthogonal and so$(\psi_{v}x_{u}-\psi_{u}x_{v}).(\phi_{v}x_{u}-\phi_{u}x_{v})=0$
Using the fundamental forms E=1=G and F=0 we get $\psi_{v}\phi_{v}+\psi_{u}\phi_{u}=0$ which after differentiating gives $\psi_{v}\sin(u)-\psi_{u}v=0$And after that i am stuck ... any help would be appreciated.

 

[mighty2000]

I understand that you are struggling with the concept of family of curves and the problem given in the image. I would like to offer some guidance and explanations that may help you tackle this problem.

Firstly, let's start with the definition of a family of curves. A family of curves is a set of curves that are related by a common equation or parameter. In this case, the family of curves is defined by the function \psi(u,v)=const, where const is a constant. This means that all curves in this family will have the same value for \psi(u,v).

Now, let's move on to the first part of the problem. The question asks you to find the geometric interpretation of the function \psi(u,v). To do this, we need to understand the role of \psi(u,v) in the fundamental forms of the surface. As you have correctly mentioned, the fundamental forms are important in differential geometry and they provide information about the local geometry of a surface. In this case, the first fundamental form is given by E=1, G=1 and F=0. This means that the surface is a flat surface with no curvature. Now, the second fundamental form is given by L=0, M=0 and N=h. This means that the surface has a constant mean curvature of h. This is where \psi(u,v) comes into play. It represents the mean curvature of the surface and hence, it can be interpreted as the height of the surface at any given point.

Moving on to the second part of the problem, you have correctly found the value of h as a constant. This is because the local parametrization is constrained to the given function. This means that for any point on the surface, the value of h will remain constant.

For the final part, we need to show that the families of curves defined by \psi(u,v)=const and \phi(u,v)=const are orthogonal. To do this, we need to show that the tangent vectors of these two families are orthogonal. As you have mentioned, the tangent vectors are given by \psi_{v}x_{u}-\psi_{u}x_{v} and \phi_{v}x_{u}-\phi_{u}x_{v}. To show that they are orthogonal, we need to take their dot product and show that it equals 0. This will involve differentiating the equations \psi(u