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I know already that conic sections (or conics) have been widely explored and many things about them are already known. I was wondering if this sort of exploration has taken place for any other sorts of solids. Spheres, of course, are easy, since each cross section is a circle. I was wondering about things like cylinders, paraboloids, or tori. I am most intrigued by the torus, especially after noticing that, when cut the right way, the cross section forms the shape of a lemniscate.

Does anyone know if the cross sections of any other solids have been investigated already?

Franklin Pezzuti Dyer
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  • I would be astounded if the cross-sectional properties of every class of three-dimensional solid haven't been fully studied, enumerated and classified. These are used in computer graphics and CAD software too, of course. – David G. Stork May 18 '17 at 23:04
  • So would I, but when I google "Toroidal Section" I don't get anything even vaguely resembling the neatly sorted and classified cross-sections of cones. – Franklin Pezzuti Dyer May 18 '17 at 23:06

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"Toric section(s)" should be a more successful search term. I've recently completed an article about toric sections, their equation, short descriptions of their most renowned types and an unexpected relation with the cone-cylinder intersections.

The toric sections: a simple introduction

Hope it helps.

Luca M
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  • Wow, thank you so much! That is very interesting and I will definitely read it. I had already identified Bernoulli's lemniscate (and its variants), but I had no idea how to classify the "tilted" sections! – Franklin Pezzuti Dyer Aug 03 '17 at 19:39
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    Your text is a pure marvel. Too few people do 3D classical analytical geometry with figures. I attempt in my answers or questions (like https://math.stackexchange.com/q/2626356 : see fig. 2 and 3) to provide pictures wherever I can. – Jean Marie Jan 14 '20 at 07:37