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Scrutinizing the General Conic Equation

Author

Listed:
  • 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)

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.

Suggested Citation

  • Mauricio Chávez-Pichardo & José Daniel López-Barrientos & Saúl Perea-Flores, 2025. "Scrutinizing the General Conic Equation," Mathematics, MDPI, vol. 13(9), pages 1-26, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:9:p:1428-:d:1643476
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