Reduction of Curves and Surfaces to Canonical Form

Reduction to Canonical Form

Algorithm for Second-Order Curves

Given: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

Step 1: Find the eigenvalues of the matrix of the quadratic part $A_q = \begin{bmatrix} A & B/2 \ B/2 & C \end{bmatrix}$. The characteristic polynomial: $\lambda^2 - (A+C)\lambda + (AC - B^2/4) = 0$.

Step 2: By rotation (along the eigenvectors), eliminate the $xy$ term. Angle: $\tan 2\varphi = \dfrac{B}{A-C}$.

Step 3: By shifting the coordinates (completing the square), eliminate the linear terms.

Step 4: By the signs of the coefficients at the squares, determine the type.

Invariants

$I_1 = A + C$ (the trace of the quadratic part). $I_2 = AC - B^2/4$ (the determinant of the quadratic part). $I_3 = \begin{vmatrix} A & B/2 & D/2 \ B/2 & C & E/2 \ D/2 & E/2 & F \end{vmatrix}$ (the full matrix).

The type of the curve is completely determined by the signs of $I_2$, $I_3$, and $I_1/I_3$ when $I_2 > 0$.

Curves in Homogeneous Coordinates

In projective space, all three types of non-degenerate conic sections are projectively equivalent. The distinction between ellipse, parabola, and hyperbola lies in their position relative to the "line at infinity" (the line $z = 0$ in homogeneous coordinates).

Applications in Computational Geometry

Finding the intersection of a ray with a second-order surface reduces to a quadratic equation (algorithms for ray tracing in computer graphics). The discriminant determines the number of intersection points.