MATHEMATICS LAB · GEOMETRY

General and rotated quadrics

Study a second-degree surface with mixed terms xy, xz and yz. Rotate the cutting plane freely to explore its sections, diagonalization and principal axes.

The coefficients are real numbers. Coordinates are given in arbitrary units (u), while the quadratic matrix uses numerical coefficients with respect to u. Unbounded surfaces are displayed only over a finite region; even oblique sections are classified analytically.

1. Define the quadric

Ten real coefficients

Ax² + By² + Cz² + Dxy + Exz + Gyz + Hx + Iy + Jz + K = 0

A · x² B · y² C · z² D · xy E · xz G · yz H · x I · y J · z K · constant

The coefficient of xy appears in the matrix as D/2; likewise for xz e yz. An orthogonal rotation eliminates the mixed terms.

Cutting plane

Plane orientation Plane offset (u) Move the plane 0 u

Examples:

3D axis rotation

In the canonical view, the quadric and the plane are progressively aligned with the XYZ axes without deformation. You can also show the reference-frame rotation alone.

Show quadric rotationDisplay the transition to canonical form (not a physical deformation). Progress 100%

Axes aligned with eigenvectors (100%).

2. Interactive 3D graph

Animated surface and reference frame

General quadricsSelect an example to display the surface.

Drag to rotate the view; use the mouse wheel or + and − to zoom. Play automatically scrolls to the graph. The slider shows the same quadric in a moving reference frame: when enabled, the surface, section and principal axes rotate together without distortion. You can also select a freely oriented plane using inclination, azimuth and offset, or a plane parallel to the principal U, V or W axes.

Surface Plane Intersection Principal axes

Equation comparison

Equation in the original frame

In the principal frame

3. How to read the construction

4. Results

Complete classification

5. Explained calculations

Matrix, eigenvalues, sections

What you can study

Classification of quadrics

Ellipsoids, hyperboloids, paraboloids, cones, cylinders and degenerate cases, including rotated and translated surfaces.

Orthogonal diagonalization

Eigenvalues, eigenvectors, a symmetric 3×3 matrix, coordinate changes and elimination of mixed terms.

Plane sections

Identify ellipses, hyperbolas, parabolas, lines, points and sections with no real intersection.

Unbounded curves and surfaces are drawn only within a finite region. The 3D surface is approximated using a triangle mesh; classifications and equations are computed algebraically. The section plane may be parallel to a Cartesian plane, to a principal plane, or have any orientation.