How this calculation works
The quadratic part distinguishes elliptic, parabolic and hyperbolic families, while additional determinant and real-point checks identify exceptions. A sampled implicit graph also handles the xy rotation term.
Ax² + Bxy + Cy² + Dx + Ey + F = 0; Δ = B² − 4ACA worked example
For x²+y²−9=0, the result is a circle of radius three centered at the origin.
Good to know
Bounded real expressions and floating-point arithmetic. Finite graph samples can miss singularities, isolated features or behavior outside the selected window. Original domain exclusions still apply.
Begin with the quadratic discriminant
Compute B²−4AC from the quadratic coefficients. A negative value describes the elliptic family, zero the parabolic boundary and a positive value the hyperbolic family. A nonzero xy term can rotate principal axes relative to the coordinate axes.
The discriminant does not by itself prove that an ordinary real curve exists. Both x²+y²=1 and x²+y²=−1 have the same negative discriminant, but only the first has real points. The tool therefore checks more than the family sign.
Recognize degenerate and empty cases
A point, a line, intersecting lines or parallel lines can arise as degeneracies of a quadratic equation. The homogeneous 3×3 coefficient matrix provides a determinant test for degeneracy. Near-zero floating-point results need exact algebra when a classification matters.
If the quadratic coefficients all vanish, the equation may describe a line, an empty set or the entire plane. The all-zero equation is rejected as a non-curve. An elliptic quadratic with an incompatible centered constant is labeled an empty real ellipse rather than drawn as an ordinary ellipse.
Read the implicit graph carefully
The contour comes from evaluating the equation on a finite square grid and interpolating edge crossings. This avoids having to solve only for y, which can omit vertical or rotated branches. The graph window is controlled by the half-width field.
Finite contours can miss isolated points, tangencies, thin features and curves outside the selected window. A blank contour is therefore not a nonexistence proof. Increase the window for distant features and use exact algebra for degenerate equations.
Connect the picture with simpler equations
Completing the square converts a translated circle or ellipse into a recognizable centered form. Rotating axes removes the xy cross term when appropriate. These transformations explain the curve rather than treating its name as a pattern-recognition trick.
The parabola vertex and axis of symmetry calculator specializes in ordinary vertical parabolas, where the vertex and axis can be read directly. The function composition calculator instead studies explicit single-input functions. An implicit conic may require two branches or may not be a single-valued function of x at all.
A few questions, answered
Can it graph a rotated ellipse?
Yes. The implicit contour accepts a nonzero xy coefficient.
Does negative discriminant always mean a real ellipse?
No. Empty and degenerate cases need additional checks.
Why is a point sometimes missing from the graph?
An isolated point may not cross any sampled grid edge.