determining if a curve passes through a loop

Question

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.

Answer 1

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. $$x_1(t_1)<x_2(t_0)<x_1(t_2)\quad\text{and}\quad y_1(t_1)<y_2(t_0)<y_1(t_2)\,\text{or}\,y_1(t_1)>y_2(t_0)>y_1(t_2)$$