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Parametrizing a curve

  1. Jan 7, 2013 #1
    Is there a general way to find a vector valued function that parametrizes a curve??? I'm reading through a textbook and it says nothing in depth about parametrization and suddenly there's a question...

    Find a vector valued function f that parametrizes the curve in the direction indicated.

    4x^2 + 9y^2 = 36





    Can someone please help me with this example and shed some light on parametrization????
     
  2. jcsd
  3. Jan 7, 2013 #2

    HallsofIvy

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    There is no general "algorithm" for finding a parameterization- it usually results from some geometric or physical insight into the problem. Here, for example, I would recognise the graph of [itex]4x^2+ 9y^2= 36[/itex] as an ellipse which, in "standard form", would be [itex]x^2/9+ y^2/4= 1[/itex]. In that form I can see that the ellipse has center (0, 0) and has vertices at (3, 0), (-3, 0), (0, 2), and (0, -2).

    And that I can think of as a "warped circle". Standard parametric equations for a circle with center at (0, 0) and radius R are [itex]x= R cos(\theta)[/itex] and [itex]y= R sin(\theta)[/itex]. Then it should be easy to see that choosing different values for "R" will give [itex]x= 3 cos(\theta)[/itex] and [itex]y= 2sin(\theta)[/itex] will give the desired ellipse.
     
  4. Jan 7, 2013 #3
    That was extremely helpful, thank you so much.

    One more thing - when looking for a vector valued function, can I just assign the unit vectors to the x and y equations so that f(t) = 3cos(t)i + 2sin(t)j, where i and j are both unit vectors and t is the parameter?
     
  5. Jan 7, 2013 #4

    HallsofIvy

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    Yes, of course, I should have said that: If x= f(t) and y= g(t) then the "vector form" is v(t)= xi+ yj= f(t)i+ g(t)j.
     
  6. Jan 11, 2013 #5

    HallsofIvy

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    By the way, in your the post it says "in the direction indicated" but there was no "direction indicated". My parameterization "goes around" the ellipse as [itex]\theta[/itex] increases- in the counterclockwise direction. If, instead we use [itex]x= 3cos(t)[/itex], [itex]y= -2sin(t)[/itex], or v(t)= 3cos(t)i-2 sin(t)j goes around the ellipse in the clockwise direction.
     
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