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Do any theorems other than Villarceau's theorem and closely related results say anything about intersections of planes with tori?

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I'm not sure about what you are looking for, but I've recently completed an article about toric sections, their equation and an unexpected relation with cone-cylinder intersections.

The toric sections: a simple introduction

Hope it helps.

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    What I had in mind was actually this: https://mathoverflow.net/questions/260202/jacobis-elliptic-functions-and-plane-sections-of-a-torus2017-08-03