Calculator inputs
Formula used
The matrix Q stores every quadratic coefficient. Off-diagonal entries represent half of each cross term. Eigenvectors rotate coordinates into principal directions.
A centre satisfies 2Qx + L = 0. Translation removes linear terms when Q is invertible. Eigenvalue signs classify the transformed quadratic surface.
How to use the calculator
- Select a three-dimensional surface or two-dimensional conic.
- Choose coefficients, equation text, or symmetric matrix input.
- Enter numeric values and optional trace information.
- Select precision and preferred angle units.
- Press Analyse equation to view the complete result.
- Use the export buttons for saved result copies.
Quadric classification guide
| Canonical pattern | Typical classification | Main indicator |
|---|---|---|
| Three same-sign squares equal one | Ellipsoid or sphere | Definite full-rank matrix |
| Two positive and one negative square | One-sheet hyperboloid | Two terms match the right side |
| One positive and two negative squares | Two-sheet hyperboloid | One term matches the right side |
| Indefinite quadratic equals zero | Elliptic cone | Mixed eigenvalue signs |
| Two squares and one linear coordinate | Elliptic or hyperbolic paraboloid | One zero eigenvalue |
| One coordinate is absent | Quadratic cylinder | Rank-two matrix without axis linearity |
| Factored quadratic planes | Intersecting or parallel planes | Degenerate rank pattern |
Example data
| Equation | Expected surface |
|---|---|
| x² + y² + z² − 25 = 0 | Sphere |
| 4x² + 9y² + 16z² − 144 = 0 | Ellipsoid |
| x² + y² − z² − 1 = 0 | One-sheet hyperboloid |
| −x² − y² + z² − 1 = 0 | Two-sheet hyperboloid |
| x² + 2y² − z = 0 | Elliptic paraboloid |
| x² − y² − z = 0 | Hyperbolic paraboloid |
Frequently asked questions
What is a quadratic surface?
It is a second-degree equation in three variables. Its graph forms a quadric surface. Matrix properties reveal its geometric type.
Why are cross terms important?
Cross terms indicate rotated principal axes. Eigenvectors determine the needed coordinate rotation. The final equation then becomes easier.
What do eigenvalues show?
Eigenvalues measure curvature along principal directions. Their signs separate bounded and unbounded shapes. Zero values indicate missing quadratic directions.
How is the centre calculated?
The centre solves 2Qx plus L equals zero. An invertible matrix gives one unique centre. Singular matrices may have no centre.
What makes a surface degenerate?
Degenerate equations collapse into simpler geometric sets. Examples include planes, lines, points, or emptiness. Rank tests detect these special cases.
Can this calculator analyse conic sections?
Yes, select the two-dimensional calculation mode. It classifies circles, ellipses, parabolas, and hyperbolas. Degenerate conics are also detected.
How are traces calculated?
A trace fixes one coordinate or plane. Substitution creates a two-dimensional quadratic equation. The resulting conic is classified automatically.
Why can a graph look incomplete?
The graph uses a finite numerical plotting range. Large surfaces may extend beyond that range. Increase the graph range control.
Are decimal results exact?
Matrix calculations use floating-point arithmetic. Display precision controls shown decimal places. Very ill-conditioned inputs may reduce accuracy.