Conic Classifier

General equation: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Discriminant: \(\Delta = B^2 - 4AC\). \(\Delta < 0\): ellipse (circle if \(A=C\) and \(B=0\)), non-degenerate. \(\Delta = 0\): parabola (no finite center). \(\Delta > 0\): hyperbola. Rotation to eliminate \(xy\): \( \displaystyle \theta = \tfrac12 \operatorname{atan2}(B, A - C) \). Quadratic form matrix \(Q=\begin{bmatrix}A & B/2\\ B/2 & C\end{bmatrix}\) has eigenvalues \(\lambda_{1,2}\). After rotation, \(B' = 0\) and the principal coefficients are \(\lambda_1,\lambda_2\). Center (for \(\Delta \neq 0\)) by solving the linear system of partials: \[ \begin{bmatrix} 2A & B \\ B & 2C \end{bmatrix} \begin{bmatrix} x_0 \\ y_0 \end{bmatrix} = -\begin{bmatrix} D \\ E \end{bmatrix}. \] Evaluate at the center: \(F' = A x_0^2 + B x_0 y_0 + C y_0^2 + Dx_0 + Ey_0 + F\). Canonical (rotated/translated) form: \(\lambda_1 u^2 + \lambda_2 v^2 + F' = 0\). Ellipse if \(\lambda_1\!>\!0,\lambda_2\!>\!0, F'\!<\!0\): \( \frac{u^2}{a^2} + \frac{v^2}{b^2} = 1\) with \( a=\sqrt{\frac{-F'}{\lambda_1}},\; b=\sqrt{\frac{-F'}{\lambda_2}}.\) Hyperbola if \(\lambda_1\lambda_2\!<\!0\): \( \operatorname{sgn}(\lambda_1)\frac{u^2}{a^2} + \operatorname{sgn}(\lambda_2)\frac{v^2}{b^2} = 1\) with \( a=\sqrt{\left|\frac{F'}{\lambda_1}\right|},\; b=\sqrt{\left|\frac{F'}{\lambda_2}\right|}.\) Degeneracy check (optional): determinant of the conic matrix \(\displaystyle \det\!\begin{bmatrix}A & B/2 & D/2\\ B/2 & C & E/2\\ D/2 & E/2 & F\end{bmatrix}=0\) indicates a degenerate conic (point, single line, or pair of lines).

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