Enter the general quadratic coefficients
Use the equation form Ax² + Bxy + Cy² + Dx + Ey + F = 0. The calculator rotates and shifts the conic into a canonical form.
Example data table
| Case | General equation coefficients | Detected form | Canonical result |
|---|---|---|---|
| Ellipse | A=9, B=0, C=4, D=-54, E=-16, F=61 | Ellipse centered at (3,2) | X²/4 + Y²/9 = 1 |
| Hyperbola | A=4, B=0, C=-9, D=-8, E=-54, F=41 | Hyperbola centered at (1,-3) | Y²/4 - X²/9 = 1 |
| Parabola | A=0, B=0, C=1, D=-4, E=0, F=0 | Right-opening parabola | y² = 4x |
| Circle | A=1, B=0, C=1, D=-6, E=8, F=9 | Circle centered at (3,-4) | X² + Y² = 16 |
Formula used
General conic: Ax² + Bxy + Cy² + Dx + Ey + F = 0
Discriminant: Δ = B² - 4AC
Rotation angle: θ = ½ atan2(B, A - C)
Principal quadratic values: λ₁, λ₂ = ((A + C) ± √((A - C)² + B²)) / 2
Center for central conics: solve [2A B; B 2C][h;k] = [-D;-E]
Translated principal form: λ₁X² + λ₂Y² + Fc = 0
The calculator first removes the xy term through rotation. Next it shifts the origin to the center or vertex. The resulting equation is then normalized into ellipse, circle, hyperbola, or parabola form whenever a real canonical form exists.
How to use this calculator
- Enter coefficients A, B, C, D, E, and F from your quadratic equation.
- Press Submit to classify the conic and compute the principal transformation.
- Review the detected conic type, canonical equation, invariants, and coordinate rules.
- Inspect the graph to verify the curve geometry and rotation visually.
- Use the CSV or PDF buttons to save the summary table for reports or classwork.
FAQs
1. What does canonical equation mean here?
It means the conic has been rotated and shifted into a simplified standard form. That form reveals the geometry directly, such as axis lengths, vertex position, or hyperbola orientation.
2. Which equations can this calculator handle?
It accepts any second-degree two-variable equation written as Ax² + Bxy + Cy² + Dx + Ey + F = 0. This includes circles, ellipses, parabolas, hyperbolas, and some degenerate cases.
3. Why is the rotation angle important?
The rotation angle removes the mixed xy term. Once that term disappears, the conic aligns with its principal axes, making the canonical form easier to interpret and normalize.
4. How does the calculator identify the conic type?
It mainly uses the discriminant B² - 4AC, then refines the result with translated constants and eigenvalues. That combination distinguishes real central conics, parabolas, imaginary forms, and degenerate cases.
5. Why might no real canonical curve appear?
Some coefficient sets represent imaginary or degenerate conics. In those cases, the algebraic equation exists, but the graph has no real locus or collapses into a point, line, or line pair.
6. Are the exported CSV and PDF files useful for homework?
Yes. They capture the submitted equation, type, angle, invariants, and canonical result. That makes it easier to attach clean computational evidence to assignments, notes, or technical reports.
7. Does the plot display rotated conics correctly?
Yes. The graph uses the original implicit equation and draws the zero contour directly. Because it plots the full equation, rotation and translation effects appear automatically on the graph.
8. Can I use decimals or negative values?
Yes. Every coefficient field accepts signed decimals. That makes the calculator suitable for classroom exercises, symbolic checks, and applied geometry problems with scaled coordinate systems.