Wikipedia:Reference desk/Archives/Mathematics/2024 February 14

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February 14

Conic sections and R ∪ {∞}

(I'm not sure of the notation) Is it reasonable to view the conic sections as always being ellipses in the two dimentional reals with a point(?) at Infinity, with the Parabola being an ellipse that touches infinity and the Hyperbola passing through Infinity and coming out the other side?Naraht (talk) 17:26, 14 February 2024 (UTC)

You need infinitely many points at infinity to turn a standard Euclidean space of dimension 2 or higher into a projective space; for the Euclidean plane this is called a "line at infinity". All non-degenerate conic sections are in some sense circles in the extended Euclidean plane. So this is reasonable if you are prepared to commit yourself to going by the rules of projective geometry. See the last paragraph of the lead section of Conic section and the section Conic section § In the real projective plane, as well as the paragraph in Projective geometry § Description that starts with "Projective geometry also includes a full theory of conic sections".  --Lambiam 17:56, 14 February 2024 (UTC)

Thank you. Figured that at some point, it would move from Euclidean to Projective, but couldn't figure out how.Naraht (talk) 03:14, 16 February 2024 (UTC)