Enter Quadric Coefficients
Example Data Table
| Surface | A | B | C | D | E | F | G | H | I | J | Action |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Unit ellipsoid | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | |
| One-sheet hyperboloid | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | |
| Elliptic cone | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| Elliptic cylinder | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | |
| Elliptic paraboloid | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
Formula Used
General quadric form: Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
Matrix form: xᵀQx + pᵀx + J = 0
Quadratic matrix:
| A | D/2 | E/2 |
| D/2 | B | F/2 |
| E/2 | F/2 | C |
Center condition: Qc = -p/2, where p = [G, H, I]ᵀ
Translated constant: K = cᵀQc + pᵀc + J
Gradient at (x, y, z):
- ∂f/∂x = 2Ax + Dy + Ez + G
- ∂f/∂y = 2By + Dx + Fz + H
- ∂f/∂z = 2Cz + Ex + Fy + I
How to Use This Calculator
- Enter the quadratic coefficients A through J from your surface equation.
- Add optional test point values if you want the equation value and gradient there.
- Click Calculate to analyze the surface below the header and above the form.
- Read the estimated surface type, matrix rank, determinant, eigenvalues, center status, and translated constant.
- Use Download CSV for data export.
- Use Save Result as PDF to print or save the report.
- Try the example table to load common engineering quadric cases quickly.
Quadric Surface Equation Calculator in Engineering
Quadric surfaces appear in geometry, design, simulation, and structural modeling. Engineers often work with ellipsoids, cones, cylinders, paraboloids, and hyperboloids. These shapes describe pressure vessels, reflectors, shells, transition parts, and coordinate transformations. A reliable quadric surface equation calculator helps reduce manual algebra and speeds up interpretation.
Why the general equation matters
The full second-degree form is flexible. It captures squared terms, mixed terms, linear terms, and a constant. That means one expression can represent many families of surfaces. When cross terms appear, the surface may be rotated. When linear terms appear, the surface may be shifted. This calculator keeps those details visible.
What the calculator analyzes
The tool converts the equation into a symmetric quadratic matrix. From that matrix, it finds the determinant, trace, principal minor sum, rank, and eigenvalues. Those values reveal the signature of the surface. They also show whether the surface is central, degenerate, or likely parabolic. This is useful in engineering math, CAD preparation, and analytic geometry review.
Center, translation, and interpretation
If a center exists, the calculator solves the translation system directly. That step removes the linear part and produces a cleaner centered form. The translated constant then helps distinguish ellipsoids, hyperboloids, cones, cylinders, and plane pairs. If no center exists, the surface is usually non-central. In many practical cases, that points to a paraboloid or a parabolic cylinder.
Engineering value
This type of analysis supports report writing and model checking. It helps students verify homework, and it helps engineers confirm whether a coefficient set matches the intended surface. The optional test point output also checks whether a coordinate lies on the surface and shows the gradient direction there. That makes the calculator useful for coursework, inspection, and quick design validation.
Frequently Asked Questions
1) What is a quadric surface equation?
A quadric surface equation is a second-degree equation in x, y, and z. It can model ellipsoids, paraboloids, cones, cylinders, hyperboloids, and several degenerate surfaces.
2) What do the mixed terms xy, xz, and yz mean?
Mixed terms usually indicate rotation or coupling between axes. They show that the surface may not align with the original coordinate directions.
3) How is the center found?
The center is found by solving Qc = -p/2. If the system has a solution, the surface is central or has a family of centers.
4) Why can a quadric have no center?
Some quadrics are non-central. Paraboloids are the common example. Their linear terms cannot be removed by a simple finite translation.
5) What does the determinant of Q tell me?
det(Q) helps show whether the quadratic part is full rank. A nonzero determinant usually means the quadric has a unique center.
6) Can this calculator detect degenerate surfaces?
Yes. It can indicate single planes, line cases, plane pairs, imaginary forms, and other degenerate outcomes from the same coefficient set.
7) What does the gradient output mean?
The gradient gives the local normal direction to the implicit surface at the chosen point. It is useful for tangency and slope interpretation.
8) How do I export the result?
Use the CSV button for a spreadsheet-friendly file. Use the PDF button to print the report or save it as a PDF file.