Calculator Input
Example Data Table
| A | B | C | D | E | F | Direction 1 | Direction 2 | λ₁ | λ₂ | Nature |
|---|---|---|---|---|---|---|---|---|---|---|
| 5 | 4 | 2 | -6 | 8 | -10 | 26.565051° | 116.565051° | 6.000000 | 1.000000 | Positive definite |
| 3 | -6 | 3 | 0 | 0 | 0 | -45.000000° | 45.000000° | 6.000000 | 0.000000 | Semidefinite |
| 2 | 5 | -1 | 1 | -2 | 4 | 29.518122° | 119.518122° | 3.549510 | -2.549510 | Indefinite |
Formula Used
The calculator reads the quadratic part of the expression
f(x, y) = Ax² + Bxy + Cy² + Dx + Ey + F
It builds the symmetric matrix
M = [ [A, B/2], [B/2, C] ]
Principal directions come from the eigenvectors of this matrix. The eigenvalues are
λ₁,₂ = (A + C ± √((A - C)² + B²)) / 2
The principal rotation angle for the first direction is
θ = 1/2 · atan2(B, A - C)
The two orthonormal principal direction vectors are
v₁ = (cos θ, sin θ), v₂ = (-sin θ, cos θ)
After rotation, the mixed term disappears, so the quadratic part becomes diagonal:
λ₁u² + λ₂v²
If a unique stationary point exists, it is found by solving
[ [2A, B], [B, 2C] ] · [x, y]ᵀ = -[D, E]ᵀ
How to Use This Calculator
- Enter the coefficients A, B, and C for the quadratic part.
- Optionally add D, E, and F to analyze the full expression.
- Set the plot range to control the graph window.
- Click Calculate Principal Directions.
- Read the two principal angles, eigenvalues, and direction vectors.
- Review the rotated coefficients to confirm the mixed term disappears.
- Use the contour graph to inspect orientation and curvature visually.
- Download the current result as CSV or PDF when needed.
8 FAQs
1) What is a principal direction?
A principal direction is an orientation where the quadratic part acts without cross-coupling. In that rotated frame, the mixed xy term disappears and curvature is measured directly by the eigenvalues.
2) Why does the mixed term matter?
The mixed term tilts the geometry. When B is non-zero, the surface or conic is not aligned with the coordinate axes. Principal directions identify the natural axes of the quadratic part.
3) What do the eigenvalues tell me?
They measure curvature along the principal directions. Positive values indicate upward curvature, negative values indicate downward curvature, and opposite signs indicate saddle behavior.
4) Why are there two direction angles?
A symmetric 2D quadratic part has two perpendicular principal directions. Once one angle is known, the second is exactly 90 degrees away.
5) When are principal directions not unique?
They are not unique when the quadratic part is isotropic, meaning A equals C and B equals zero. In that case, every orthonormal pair behaves the same.
6) Does D, E, and F affect direction angles?
No. Linear and constant terms shift the graph and change the stationary point, but the principal directions depend only on the quadratic coefficients A, B, and C.
7) What does positive definite mean here?
It means both eigenvalues are positive. The quadratic part curves upward in every direction, and any unique stationary point is a minimum.
8) What does the graph represent?
The graph is a contour map of the full quadratic expression. The overlaid guide lines show the principal directions, making orientation and anisotropy easy to inspect visually.