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

r == Cos[n θ + k t] 


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.


1 Answer 1


Would this plot suffice?

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

enter image description here

  • $\begingroup$ facepalm Yes. Can't believe I didn't think of that. $\endgroup$ Jul 13, 2017 at 5:12
  • 3
    $\begingroup$ Please, could you explain how to get to this equation? $\endgroup$ Jul 15, 2017 at 7:14
  • 1
    $\begingroup$ @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. $\endgroup$
    – T.F
    Jul 23, 2017 at 13:56

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