Scrutinizing the General Conic Equation

Mauricio Chávez-Pichardo, José Daniel López-Barrientos () and Saúl Perea-Flores
Additional contact information
Mauricio Chávez-Pichardo: 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
José Daniel López-Barrientos: Facultad de Ciencias Actuariales, Universidad Anáhuac Mexico, Naucalpan de Juárez 52786, Mexico
Saúl Perea-Flores: PNC Financial Services Group, Texas Division, Dallas, TX 75206, USA

Mathematics, 2025, vol. 13, issue 9, 1-26

Abstract: We present a general formula that transforms any conic of the form A x 2 + B x y + C y 2 + D x + E y + 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 x y , 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/9/1428/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/9/1428/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:9:p:1428-:d:1643476

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().