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Mathematics
Volume 13
Issue 9
10.3390/math13091428
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Scrutinizing the General Conic Equation

by
Mauricio Chávez-Pichardo
1,
José Daniel López-Barrientos
2,* and
Saúl Perea-Flores
3
1
TecNM—Tecnológico de Estudios Superiores del Oriente del Estado de México, División de Estudios de Posgrado e Investigación y División de Ingeniería en Energías Renovables, La Paz 56400, Mexico
2
Facultad de Ciencias Actuariales, Universidad Anáhuac Mexico, Naucalpan de Juárez 52786, Mexico
3
PNC Financial Services Group, Texas Division, Dallas, TX 75206, USA
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(9), 1428; https://doi.org/10.3390/math13091428 (registering DOI)
Submission received: 14 March 2025 / Revised: 23 April 2025 / Accepted: 25 April 2025 / Published: 26 April 2025
(This article belongs to the Section B: Geometry and Topology)
Versions Notes

Abstract

We present a general formula that transforms any conic of the form Ax2+Bxy+Cy2+Dx+Ey+F=0, with B≠0, into A′(x′)2+C′(y′)2+D′x′+E′y′+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.
Keywords: analytic geometry; general second-degree equation; rotated conic sections; rotation of the Cartesian axes; degenerate and imaginary conic sections analytic geometry; general second-degree equation; rotated conic sections; rotation of the Cartesian axes; degenerate and imaginary conic sections

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MDPI and ACS Style

Chávez-Pichardo, M.; López-Barrientos, J.D.; Perea-Flores, S. Scrutinizing the General Conic Equation. Mathematics 2025, 13, 1428. https://doi.org/10.3390/math13091428

AMA Style

Chávez-Pichardo M, López-Barrientos JD, Perea-Flores S. Scrutinizing the General Conic Equation. Mathematics. 2025; 13(9):1428. https://doi.org/10.3390/math13091428

Chicago/Turabian Style

Chávez-Pichardo, Mauricio, José Daniel López-Barrientos, and Saúl Perea-Flores. 2025. "Scrutinizing the General Conic Equation" Mathematics 13, no. 9: 1428. https://doi.org/10.3390/math13091428

APA Style

Chávez-Pichardo, M., López-Barrientos, J. D., & Perea-Flores, S. (2025). Scrutinizing the General Conic Equation. Mathematics, 13(9), 1428. https://doi.org/10.3390/math13091428

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