Model general quadric equations for engineering analysis and geometry review. Solve invariants, classify surfaces, and inspect center shifts with confidence today.
Use the general form: Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
| Surface Example | A | B | C | D | E | F | G | H | I | J |
|---|---|---|---|---|---|---|---|---|---|---|
| Unit Ellipsoid | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| One Sheet Type | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| Two Sheets Type | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| Shifted Ellipsoid | 1 | 1 | 1 | 0 | 0 | 0 | -4 | -6 | -8 | 20 |
The calculator uses the quadratic surface equation:
Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
The symmetric quadratic matrix is:
Q = [[A, D/2, E/2], [D/2, B, F/2], [E/2, F/2, C]]
The linear vector is:
L = [G, H, I]
The center is found by solving:
Q · [x₀, y₀, z₀]ᵀ = -L/2
The trace, determinant, and eigenvalues of Q help identify the quadric family. After translation to the center, the shifted constant determines whether the real surface is closed, open, conical, or nonreal.
Quadratic surfaces appear in many engineering models. They describe tanks, shells, reflectors, stress boundaries, and coordinate transformations. A general quadric equation can look difficult at first. However, its matrix form reveals the real geometry. This calculator helps engineers inspect that structure quickly.
The tool accepts squared terms, mixed terms, linear terms, and one constant. It converts them into a symmetric quadratic matrix. From that matrix, it computes the trace, determinant, second invariant, and eigenvalues. These values show how the surface bends in three dimensions.
Classification depends on the signs of the eigenvalues and the translated constant. When all principal directions have the same sign, the surface usually behaves like an ellipsoid family. When one sign differs, the result may be a hyperboloid or a cone. Singular matrices often indicate parabolic or degenerate forms.
Many engineering equations include linear terms because the surface is shifted away from the origin. Solving for the center removes that offset. This translated form makes interpretation easier. It also helps when comparing design shapes, analyzing symmetry, or preparing documentation for simulation work.
Quadratic surfaces support optics, structural design, heat transfer studies, and geometric modeling. Engineers use them to describe reflectors, pressure vessel sections, and analytical boundaries in CAD or finite element workflows. A fast calculator reduces manual algebra and helps verify whether a design equation matches the intended shape.
This page also supports result export. CSV output is useful for reports, spreadsheets, and repeated checks. PDF saving helps share clean summaries with teams. Because the form uses a responsive grid, entering many coefficients remains manageable on large screens and mobile devices.
A quadratic surface is a three-dimensional shape defined by a second-degree equation in x, y, and z. Common examples include ellipsoids, cones, paraboloids, and hyperboloids.
Mixed terms such as xy, xz, and yz indicate axis coupling or rotation. They often mean the surface is tilted relative to the standard coordinate axes.
The determinant of the quadratic matrix helps detect singular behavior. A near-zero determinant often suggests a degenerate surface or a parabolic family rather than a fully centered quadric.
The center removes linear offsets from the original equation. This makes the surface easier to interpret and helps show whether the geometry is centered, shifted, or degenerate.
It gives a strong engineering classification based on invariants, eigenvalues, and the translated constant. Very special degenerate cases may still need detailed manual review.
A singular matrix means the center may not be uniquely defined. In that case, the surface is often parabolic, cylindrical, or otherwise degenerate.
Yes. It helps verify equation structure before entering geometry into CAD, numerical models, or technical reports. It is especially useful during early equation checking.
Use CSV when you want spreadsheet-ready values. Use the PDF option when you need a simple print-style record for review, sharing, or documentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.