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Parameterized function crosses own path

  1. Feb 27, 2013 #1
    Hi!

    Given a function [tex] r:\mathbb{R} \rightarrow \mathbb{R}^2, r(t) = (f_1(t), f_2(t))[/tex], is there a way to analytically determine if there are points (x1, x2) where r(t) = (x1, x2) for multiple t-values?

    Lets say i was to find such points for the function [tex] r(t) = (t^3-t, 3t^2 + 1) [/tex]

    How should i go about finding the points without having to plot the graf?

    Thanks!
     
  2. jcsd
  3. Feb 27, 2013 #2

    CompuChip

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    In theory, you could assume that given some time t, the graph crosses the same point r(t) after time a again, leading to the equation
    r(t) = r(t - a).

    This leads to two equations in t and a.
    The question would then be if there is a non-trivial solution (i.e. [itex]a \neq 0[/itex]).
     
  4. Feb 27, 2013 #3
    Exactly!

    Thanks alot :)
     
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