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Keywords = degenerate and imaginary conic sections

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26 pages, 10308 KB  
Article
Scrutinizing the General Conic Equation
by Mauricio Chávez-Pichardo, José Daniel López-Barrientos and Saúl Perea-Flores
Mathematics 2025, 13(9), 1428; https://doi.org/10.3390/math13091428 - 26 Apr 2025
Viewed by 774
Abstract
We present a general formula that transforms any conic of the form Ax2+Bxy+Cy2+Dx+Ey+F=0, with B0, into [...] Read more.
We present a general formula that transforms any conic of the form Ax2+Bxy+Cy2+Dx+Ey+F=0, with B0, into A(x)2+C(y)2+Dx+Ey+F=0, without requiring the rotation angle θ. This directly eliminates the cross term xy, simplifying the rotated conics analysis. As consequences, we obtain new formulae that remove both rotations and translations, a novel proof of the discriminant criterion, improved expressions for eccentricity, and a detailed taxonomy of all loci described by the general conic equation. Full article
(This article belongs to the Section B: Geometry and Topology)
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