Difficulties with parametric plots

FYI: This is the kind of things I am hoping to learn to produce:

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Not sure about the policy on replying to onesself here(?), but after a private conversation, I felt I should clarify this.

What I mean to say is, I find it easier to tell their api what I want explicitly; than I do to work out what I have to tell Plots api in order for it to guess correctly about what I want.

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