# Math Help - Self intersection

1. ## Self intersection

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

γ(t)=(t²-1.t³-t)

2. Let $a,b$ be the times where the curve intersects with $a \ne b$then
$x=a^2-1$ 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)

3. thanks thats nice and easy to understand, my teacher had confused me with his explanation.

4. 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.