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What are eccentricities of "Conic" section created by intersection of a one sheet hyperboloid ( meridian $e >1 $) with a plane at inclination $\phi?$

Which inclination results in a hyperbola? a parabola? Thanks in advance.

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Narasimham
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  • https://en.wikipedia.org/wiki/Hyperboloid#Plane_sections – Intelligenti pauca Aug 04 '21 at 14:36
  • Thank you. The above Wiki has answers I wanted at one place... almost matching with expectation and intuition. Please answer if possible including how circular sections may be possible on a 1- sheet hyperboloid; to me it ( barring the central obvious case) looked like counter-intuitive. – Narasimham Aug 04 '21 at 14:58
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    Start with a horizontal plane and rotate it about the major axis of the elliptic intersection: for some angle the minor axis will have the same length as the major one, thus giving a circle. – Intelligenti pauca Aug 04 '21 at 15:30
  • See also here: https://en.wikipedia.org/wiki/Circular_section#Elliptical_hyperboloid_of_one_sheet – Intelligenti pauca Aug 04 '21 at 15:41

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I don't have the answers but I think you can find them in the book New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian, 2012.

or here

New Descriptions of Conics via Twisted Cylinders, Focal Disks, and Directors (same authors as above) https://www.tandfonline.com/doi/abs/10.1080/00029890.2008.11920594

Euclid Looked On Beauty Bare
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