# Trace of the points of intersection of two moving curves

I would like to trace the points of intersection between the polar curves

r == Cos[n θ + k t]


and

r == t Csc[θ]


as t goes from -1 to 1. (n and k are constants, and y, r, and θ relate in the usual way.)

Note the the 2nd is equivalent to y == t.

Here is a GIF (taken from this article) to demonstrate what I mean:

(I really only need the equivalent of the final frame.)

Since it is not possible to obtain a closed-form solution, one would have to find the intersections numerically. While it is possible to sample a bunch of points, doing so would result in just that: a bunch of points, when what I was looking for was something like an InterpolatingFunction.

Would this plot suffice?

ContourPlot[
Sqrt[x^2 + y^2] == Cos[5 ArcTan[x,y] + 17 y - 1],
{x, -1, 1},
{y, -1, 1},
PlotPoints->100
]


• facepalm Yes. Can't believe I didn't think of that. Jul 13, 2017 at 5:12
• Please, could you explain how to get to this equation? Jul 15, 2017 at 7:14
• @pH13, r=Sqrt[x^2 + y^2] and θ=ArcTan[x,y]. He's just making both equations equal to each other and using ContourPlot to find the intersection.
– T.F
Jul 23, 2017 at 13:56