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What numerical procedure is be adopted to detect self-intersecting parametrized points $ [x(t), y(t) ] $ in $ \mathbb R^2 $ ?

Observation : @ roots ( t= 2, t=-1 ) parabola has double value with respect to cubic. How to build an algorithm ?

Self Intrxn Parm_Graph

Narasimham
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  • an explicit example in your mind? – janmarqz Dec 25 '14 at 19:36
  • Seeing the post here last year for $ x=t^2−t, y=t^3−3t−1 $, it strongly occurs (to me) there should be a geometrical/graphical/ equation based way somewhat like Newton Raphson root finding,to get the self intersection point. I'll upload separate x,y plots soon. – Narasimham Dec 25 '14 at 20:12
  • Referring to http://math.stackexchange.com/questions/338480/find-the-point-where-equations-x-t2-t-and-y-t3-3t-1-cross-itself?rq=1 – Narasimham Dec 25 '14 at 20:26
  • It might help if you state clearly that you are looking for an algorithm (similar to Newton's method) for finding, numerically, points of self-intersection. Adding tags "algorithms" and/or "numerical-methods" might also help. – Moishe Kohan Dec 25 '14 at 21:12

1 Answers1

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I think the way to find out is that: $\exists t_1 \neq t_2: (x(t_1),y(t_1)) = (x(t_2),y(t_2))$. For if $r(t) = (x(t),y(t))$ is a continuously differentiable function on $\mathbb{R}$, then if $r'(t) = 0$ for some $t$ then you have a self intersection point.

Example: $r(t) = (t^2-t, t^3-t), t \in \mathbb{R}$.

We have $r(0) = (0,0) = r(1)$.

DeepSea
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