Scrutinizing the General Conic Equation

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.