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Math Help - Self intersection

  1. #1
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    Self intersection

    Show that the following curve is not closed and that it has exactly one self-
    intersection

    γ(t)=(t-1.t-t)
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  2. #2
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    Let a,b be the times where the curve intersects with a \ne bthen
    x=a^2-1 and y=a^3-a and
    x=b^2-1 and y=b^3-b

    Subtracting the x equations gives
    0=a^2-b^2 \implies a=\pm b
    Since a\ne b \implies a=-b

    Plugging into the y equations and subtracting gives
    -2b^3+2b=0 \implies b=0,\pm 1
    So the curve intersects itself when t=-1 and t=1 at the point (0,0)
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  3. #3
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    thanks thats nice and easy to understand, my teacher had confused me with his explanation.
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  4. #4
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    Also, note that as t goes to infinity, both t^2-1 and t^3- 1 go to infinity but as t goes to negative infinity, [tex]t^2- 1[tex] goes to infinity while t^3- 1 goes to negative infinity. That shows that the path is not closed.
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