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Parametric representation of curves

  1. Jul 29, 2010 #1
    1. The problem statement, all variables and given/known data

    See figure.


    2. Relevant equations

    N/A


    3. The attempt at a solution
    I've dealt with parametric equations for lines before in my linear algebra class but I'm not sure how I'm suppose to model two curves with one.

    Anyone have any suggestions?
     

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  3. Jul 29, 2010 #2
    Those equations are two quadratic surfaces (I'm assuming we are in 3D).
    you can try to combine them into one y=f(x) equation witch will give you a curve (assuming you want the curve of the intersection).
    Moreover, if you need, you can model it with an independent variable (t) :)
     
  4. Jul 29, 2010 #3
    I'm still pretty confused on how I'm suppose to get parametric equations from this. I could rewrite the two equations in one as,

    [tex]1 = x^{2}-xy+y[/tex]

    but I don't see how that helps me...:confused:
     
  5. Jul 29, 2010 #4
    [tex] z = 1 - xy [/tex]

    [tex] 1 = x^2 - xy + y \ \Rightarrow \ y = \frac{1 - x^2}{1 - x} = 1 + x [/tex]

    So we can rewrite z as

    [tex] z = 1 - x(1 + x) [/tex].

    Now can you think of a good variable to use for the parameterization from here?

    Hint: it's extremely simple ^^
     
  6. Jul 29, 2010 #5
    I'm not really sure... Maybe,

    [tex]x=t[/tex]?

    If so then,

    [tex]x(t) = t[/tex]

    [tex]y(t) = 1 + t[/tex]

    [tex]z(t)=1 - t(1+t)[/tex]

    I'm used to seeing parametric equations in the following form(straight lines),

    [tex]z = A + Bt[/tex] where [tex]A&B\in Z[/tex]

    So this is sorta new for me :blushing:
     
    Last edited: Jul 29, 2010
  7. Jul 29, 2010 #6
    yep, x=t works well! The way to write it is...

    x(t) = ... \
    y(t) = ... |- t [tex] \in \textbf{R}[/tex]
    z(t) = ... /

    ...where that thing in the middle is supposed to be a curly bracket...I'm too lazy to make it work in LaTeX haha
     
  8. Jul 29, 2010 #7
    It's all good! I edited my post above with the equations of x(t), y(t), and z(t).

    Thanks again for your help!
     
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