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Family of surfaces (Diff Geometry)

  1. May 11, 2008 #1
    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.

    1. The problem statement, all variables and given/known data

    (this link has an image of the problem)

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

    3. 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:
    [itex] xz-hy=> v*sin(u)=h(1-cos(u))=> h= v*sin(u)/(1-cos(u))[/itex] which is a constant.

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

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