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I have two curves described by parametric equations, and one is a closed loop. How do I analytically determine whether or not the other curve passes through the loop? That is, without graphing and visually inspection the graph. Also assuming that both paths are smooth, continuous, and have no singularities.

rurouniwallace
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1 Answers1

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If the two curves are defined by $$\gamma_1(t)=(x_1(t),y_1(t))\quad\text{and}\quad\gamma_2(t)=(x_2(t),y_2(t))$$ so you have to verify that there's $t_0,t_1,t_2$ s.t. $$x_1(t_1)<x_2(t_0)<x_1(t_2)\quad\text{and}\quad y_1(t_1)<y_2(t_0)<y_1(t_2)\,\text{or}\,y_1(t_1)>y_2(t_0)>y_1(t_2)$$

  • This would determine if the paths cross each other, but I'm looking for something to determine if the curve passes through the space surrounded by the loop. – rurouniwallace Apr 26 '13 at 00:39
  • If the second curve (A) intersects the loop one (B), then certainly A enters the loop area. Is that what you mean by "passes through". – bubba Apr 26 '13 at 01:24
  • Could the two curves be anything whatsoever, or are we dealing with certain specific types of curves (lines, circles, Bezier curves, ...)? – bubba Apr 26 '13 at 01:25
  • @bubba Yes, these curves could be anything at all. – rurouniwallace May 09 '13 at 21:11
  • OK. But you still have to properly define "passes through". Do you mean just "enters" (one end inside), or do you mean "enters and leaves too" (has intersections, but both ends outside). – bubba May 10 '13 at 01:09