conic classifier
Enter the coefficients of a general second-degree equation and instantly classify the conic (ellipse/circle, parabola, hyperbola, or degenerate). The tool also computes the rotation angle to remove the \(xy\) term, the center (when it exists), canonical form, and (semi-)axes. Ideal for engineers, surveyors, and students verifying analytic geometry work.
Interactive calculator
Coefficient of \(x^2\) in \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
Coefficient of \(xy\). If nonzero, the axes are rotated; the tool reports the angle \(\theta\) to eliminate \(xy\).
Coefficient of \(y^2\).
Coefficient of \(x\).
Coefficient of \(y\).
Constant term.
Results
Classification
—
Discriminant Δ = —
Degeneracy test det(𝙲) = —
Rotation to remove xy
θ = — (deg)
Formula \( \theta = \tfrac12 \operatorname{atan2}(B, A - C) \)
Center (if exists)
—
Principal coefficients
λ₁ = —, λ₂ = —
Canonical form (rotated/translated)
Oriented axes preview (not to scale)
Blue = principal axes; red dot = computed center (if defined).
Data source and methodology
AuthoritativeDataSource: Larson & Edwards, Calculus, Appendix D: “Rotation and the General Second-Degree Equation,” Cengage (public appendix PDF). “Ax² + Bxy + Cy² + Dx + Ey + F is classified by the discriminant: ellipse/circle if \(B^2-4AC<0\), parabola if \(=0\), hyperbola if \(>0\); rotation by \(\theta=\tfrac12\operatorname{atan2}(B, A-C)\) eliminates \(xy\).” Direct link: Appendix D (PDF). All computations strictly follow these formulas.
Supplementary: NTNU lecture notes on conics (classification by discriminant and degeneracy notes): Introduction to Conic Sections (PDF). “All calculations are rigorously based on the formulas and data provided by these sources.”
The formula explained
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).
Glossary of variables
- A,B,C,D,E,F: coefficients of the general equation.
- Δ: discriminant \(B^2-4AC\) (type test).
- θ: rotation angle (degrees) to remove \(xy\).
- (x₀,y₀): center (for Δ≠0), solution of the partial-derivative system.
- λ₁, λ₂: eigenvalues of \(Q\); principal quadratic coefficients after rotation.
- F′: constant term after translation to the center.
- a,b: semi-axes (ellipse) or axis parameters (hyperbola) in canonical form.
How it works: a step-by-step example
Example: ellipse
Use \(A=4, B=2, C=5, D=0, E=0, F=-20\). Then \(\Delta=2^2-4\cdot 4\cdot 5=-76<0\Rightarrow\) ellipse. The rotation angle is \(\theta=\tfrac12\operatorname{atan2}(2,4-5)\approx -31.72^\circ\). The center solves \(\bigl[\!\begin{smallmatrix}8&2\\2&10\end{smallmatrix}\!\bigr]\bigl[\!\begin{smallmatrix}x_0\\y_0\end{smallmatrix}\!\bigr]=\bigl[\!\begin{smallmatrix}0\\0\end{smallmatrix}\!\bigr]\Rightarrow (0,0)\). With \(Q=\bigl[\!\begin{smallmatrix}4&1\\1&5\end{smallmatrix}\!\bigr]\), \(\lambda_1\approx 3.382\), \(\lambda_2\approx 5.618\). Since \(F'=-20\), the canonical form is \(\lambda_1 u^2+\lambda_2 v^2-20=0\Rightarrow\frac{u^2}{20/\lambda_1}+\frac{v^2}{20/\lambda_2}=1\) with \(a\approx 2.43,\; b\approx 1.89\).
FAQ
How do you tell circle vs ellipse?
Both have \(\Delta<0\). A circle additionally satisfies \(A=C\) and \(B=0\).
What does “degenerate” mean here?
The equation represents a point, a single line, or two lines (parallel or intersecting) instead of a proper conic. We flag this when the 3×3 conic determinant is ~0.
Why doesn’t a parabola have a center?
Parabolas are not centrally symmetric; the linear system for \((x_0,y_0)\) becomes singular when \(\Delta=0\).
What if the discriminant test and determinant disagree?
The discriminant decides the family (ellipse/parabola/hyperbola) under non-degeneracy. A near-zero 3×3 determinant indicates a degenerate case within that family.
Do floating-point round-off errors affect results?
They can. We treat values with \(|v|<10^{-10}\) as zero for stability. Slightly perturb inputs or increase precision if you see borderline classifications.
Can I recover the focal length of a parabola here?
This tool focuses on classification, angle, and canonical coefficients. For parabolas, focal parameters require an additional completion-of-squares step after rotation; planned for a future pro version.
Is the SVG drawing exact?
No—it's a qualitative orientation preview (principal axes and center). Use the numeric outputs for precise work.
Tool developed by Ugo Candido. Content verified by CalcDomain Editorial Board.
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