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Parametric equation of a surface-eliminating the parameters

  1. Sep 28, 2009 #1
    1. The problem statement, all variables and given/known data
    The parametric equation of a surface is given by:
    x = s + t, y = (s2/2) + s f(t) + t, u = s + f(t) where f is some given function (e.g. f(t)=t/2).
    I would like to eliminate s and t from these equations and describe the surface in terms of x, y and u only.

    2. Relevant equations
    N/A

    3. The attempt at a solution
    The first equation implies that s=x-t
    Put it into the third equation, I get
    u = x - t + f(t)
    => u - x = f(t) - t
    Now if I can solve for t in terms of u and x, then I can put it into the second equation and I'm done.
    But how can I solve for t in the equation u - x = f(t) - t ?
    I think this may be related in some way to the idea of "inverse function", but I don't know how to handle it in this case...

    Can someone please help me out?
    Thank you! :)
     
  2. jcsd
  3. Sep 28, 2009 #2

    tiny-tim

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    Hi kingwinner! :smile:

    If f(t) = t/2, it's easy!

    If f(t) = t2, you need to solve a quadratic equation.

    If f(t) is more complicated, there may be no non-computer solution.
     
  4. Sep 28, 2009 #3
    Hi tiny-tim,

    If we assume that the general function f is invertible, can we eliminate s and t from these equations and describe the surface in terms of x, y and u only (perhaps with f and f-1 in the expression) ?
     
  5. Sep 29, 2009 #4

    tiny-tim

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    All functions are invertible (at least locally) …

    no, there's no general solution, you have to work it out for each one.
     
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