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Inspired by the fantastic answers herehere and herehere, I would like to ask a questions along similar lines.

Here is a start, using @ybeltukov 's spline method from herehere.

Inspired by the fantastic answers here and here, I would like to ask a questions along similar lines.

Here is a start, using @ybeltukov 's spline method from here.

Inspired by the fantastic answers here and here, I would like to ask a questions along similar lines.

Here is a start, using @ybeltukov 's spline method from here.

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I posted this image on a questionquestion on physics SE and John RennieJohn Rennie gave a link to an interesting PDF here on a similar subject.

I posted this image on a question on physics SE and John Rennie gave a link to an interesting PDF here on a similar subject.

I posted this image on a question on physics SE and John Rennie gave a link to an interesting PDF here on a similar subject.

    Notice removed Draw attention by martin
    Bounty Ended with Alexey Bobrick's answer chosen by martin
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Further update

Below are some images overlaying the curves in OP over @Alexey Bobrick 's excellent answer:

enter image description here

They approach the solution, but are missing some of the parameters:

  • The curves must pass through the origin if possible.
  • The curves should be clockwise from point 1 to point 2.
  • If a loop occurs on $y=0$, then the intersection of the loop should pass through the origin.

I believe that if these parameters are met, this approach may well provide a solution to the problem.

Further update

Below are some images overlaying the curves in OP over @Alexey Bobrick 's excellent answer:

enter image description here

They approach the solution, but are missing some of the parameters:

  • The curves must pass through the origin if possible.
  • The curves should be clockwise from point 1 to point 2.
  • If a loop occurs on $y=0$, then the intersection of the loop should pass through the origin.

I believe that if these parameters are met, this approach may well provide a solution to the problem.

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