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Thread: 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 $\displaystyle a,b$ be the times where the curve intersects with $\displaystyle a \ne b$then
    $\displaystyle x=a^2-1$ and $\displaystyle y=a^3-a$ and
    $\displaystyle x=b^2-1$ and $\displaystyle y=b^3-b$

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

    Plugging into the y equations and subtracting gives
    $\displaystyle -2b^3+2b=0 \implies b=0,\pm 1$
    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 $\displaystyle t^2-1$ and $\displaystyle t^3- 1$ go to infinity but as t goes to negative infinity, [tex]t^2- 1[tex] goes to infinity while $\displaystyle t^3- 1$ goes to negative infinity. That shows that the path is not closed.
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