TY - JOUR
T1 - Scrutinizing the General Conic Equation
AU - Chávez-Pichardo, Mauricio
AU - López-Barrientos, José Daniel
AU - Perea-Flores, Saúl
N1 - Publisher Copyright:
© 2025 by the authors.
PY - 2025/5/1
Y1 - 2025/5/1
N2 - We present a general formula that transforms any conic of the form (Formula presented.), with (Formula presented.), into (Formula presented.), without requiring the rotation angle (Formula presented.). This directly eliminates the cross term (Formula presented.), 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.
AB - We present a general formula that transforms any conic of the form (Formula presented.), with (Formula presented.), into (Formula presented.), without requiring the rotation angle (Formula presented.). This directly eliminates the cross term (Formula presented.), 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.
KW - analytic geometry
KW - degenerate and imaginary conic sections
KW - general second-degree equation
KW - rotated conic sections
KW - rotation of the Cartesian axes
UR - http://www.scopus.com/inward/record.url?scp=105005092337&partnerID=8YFLogxK
U2 - 10.3390/math13091428
DO - 10.3390/math13091428
M3 - Artículo
AN - SCOPUS:105005092337
SN - 2227-7390
VL - 13
JO - Mathematics
JF - Mathematics
IS - 9
M1 - 1428
ER -