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Parameterizing an equation?

  1. Feb 17, 2012 #1


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    How do you take an equation and turn it into a parametric one? Eliminating the parameter is straightforward and easy; but I'm trying to go the other way.

    e.g. what steps would one take to convert some curve like

    y = x2


    x2 + y2 = 1

    into parametric equations with 't' as the independent variable?
  2. jcsd
  3. Feb 17, 2012 #2
    There are number of ways, for conics, some traditional ways are:
    For y=x2; x=t, y=t2.
    For x2+y2=1; x=cost, y=sint.
  4. Feb 17, 2012 #3


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    What was your motivation for setting x = cos(t) and y = sin(t)? Is there a method for turning an equation in to a set of parametric equations?
  5. Feb 17, 2012 #4
    actually, eliminating the parameter is equally hard. If the equation is in an explicit form [itex]y = f(x)[/itex], then, whatever you take as a parametric representation of x, [itex]x = \phi(t)[/itex], you can find [itex]y = y \left[ \phi(t) \right] = \psi(t)[/itex]. In other cases, there is no general rule. For example, eliminate the parameter in:
    x = t \, \cos t, \ y = t \, \sin t
    describing an Archimedian spiral.
  6. Feb 18, 2012 #5


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    You hopefully have learned that [itex]cos^2(t)+ sin^2(t)= 1[/itex]. Comparing that to [itex]x^2+ y^2= 1[/itex] should make the parameterization obvious. But there is no general way of parameterizing (except that if y= f(x), you can always use x= t, y= f(t)).
  7. Feb 18, 2012 #6


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    Science Advisor

    Hey Fuz.

    On top of what the other posters have said, it does help immensely if you know the dimension of the system.

    If you are dealing with a one-dimensional system (like a line), then there are techniques that you can do to make a move towards getting a complete analytic parametrization.
  8. Feb 18, 2012 #7


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    Yes I have learned this, and that basically answered my question, but then what is the point in just setting x equal to t?
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