MATHEMATICS LAB · GEOMETRY

General and Rotated Conics

Enter the six coefficients of a second-degree equation. Explore the conic, rotate the axes and find its canonical form, including directrices, foci, vertices and every calculation step.

Classification uses the quadratic-form matrix and the mixed term Bxy. Even when the discriminant indicates a conic family, the real locus may be degenerate or empty. Coordinates are given in arbitrary units (u).

1. Define the conic

Six real coefficients

Examples:

Rotation of axes

The slider rotates only the reference frame: the curve remains stationary. At 100%, the mixed term disappears.

Rotation progress 100%

Principal axes in their final position.

2. Interactive conic graph

Original and principal axes

Point P on the conic (move with the slider)—

Focus used in the construction

Move P to compare the two distances.

Original equation and canonical form

Principal axes aligned

In the original frame (x, y)

In the moving frame (ξ, η)

Move the rotation slider through 0%, 50% and 100%: the curve stays fixed while the axes, coefficients, and finally the reference-frame origin change.

Drag the plot to pan; use the mouse wheel or + and − to zoom. Coefficients and axis rotation update without reloading the page.

Conic Principal axis ξ Axis η Foci, center and vertex

3. How to read the construction

4. Results

Complete classification

5. Explained calculations

Matrices and canonical form

What you can study

Complete classification

Discriminant, rank, symmetric matrix and degenerate cases: component lines, intersections and distances.

Rotation and translation

Diagonalization of the quadratic form, principal axes, center or vertex, and elimination of the mixed term.

Canonical form

Semi-axes, eccentricity, foci, directrices, tangents and normals at a point of the curve, with explained formulas.

Classification uses floating-point arithmetic and numerical tolerances; results near degenerate conics or with extremely unbalanced coefficients need caution. Only part of unbounded curves is visible. The illustrated rotation changes the coordinate system, not the original equation.