Conic Sections Calculator

The classification of a conic section depends on the discriminant:

$$\Delta = B^2 - 4AC$$

• $$\Delta < 0$$: Ellipse (or circle if $$A = C$$ and $$B = 0$$)
• $$\Delta = 0$$: Parabola
• $$\Delta > 0$$: Hyperbola

When the cross-product term $$Bxy$$ is present, the conic axes are rotated relative to the coordinate axes. The rotation angle needed to eliminate the $$xy$$ term is:

$$\phi = \frac{1}{2}\arctan\!\left(\frac{B}{A - C}\right)$$

After rotation, the coefficients transform as $$A' = A\cos^2\phi + B\cos\phi\sin\phi + C\sin^2\phi$$ and $$C' = A\sin^2\phi - B\cos\phi\sin\phi + C\cos^2\phi$$, yielding a standard-form equation without the cross term.

The eccentricity is computed from the rotated coefficients: for an ellipse, $$e = \sqrt{1 - a_{\min}/a_{\max}}$$ where $$a_{\min}$$ and $$a_{\max}$$ are the smaller and larger of the rotated coefficients. A circle has $$e = 0$$, a parabola has $$e = 1$$, and a hyperbola has $$e > 1$$.

The discriminant is the primary classifier. A negative value indicates an elliptical curve (bounded), zero indicates a parabola (one branch extending to infinity), and positive indicates a hyperbola (two branches). The conic type output uses numeric codes: 1 for ellipse, 2 for parabola, 3 for hyperbola, and 4 for circle.

Eccentricity measures how far the conic deviates from a circle: $$e = 0$$ is a perfect circle, $$0 < e < 1$$ is an ellipse, $$e = 1$$ is a parabola, and $$e > 1$$ is a hyperbola. Higher eccentricity means a more elongated or open curve.

The rotation angle indicates how much the principal axes are tilted from the standard x-y axes. If $$B = 0$$, the rotation angle is $$0°$$ and the conic is already axis-aligned.