General Second-Degree Equation & Classification of Conic Sections

1. The Universal Quadratic Form

The most general algebraic equation of the second degree in two variables is: $$F(x, y) = a x^2 + 2h xy + b y^2 + 2g x + 2f y + c = 0$$ where $a, h, b, g, f, c \in \mathbb{R}$ and $(a, h, b) \ne (0, 0, 0)$. In matrix notation, this equation can be expressed compactly using homogeneous coordinates $\mathbf{x} = (x, y, 1)^T$: $$\mathbf{x}^T \mathbf{A} \mathbf{x} = 0, \qquad \mathbf{A} = \begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix}$$

2. The Fundamental Invariants Under Rigid Euclidean Motion

Under any rigid coordinate transformation (arbitrary translation and rotation $\mathbf{x} \mapsto \mathbf{R}\mathbf{x} + \mathbf{t}$), the coefficients of the quadratic curve change, but three algebraic quantities remain strictly invariant:

1. First Invariant (Trace of Quadratic Part): $$I_1 \equiv a + b = \operatorname{tr}(\mathbf{A}_{2 \times 2})$$
2. Second Invariant (Discriminant of Quadratic Part): $$I_2 \equiv D \equiv ab - h^2 = \det(\mathbf{A}_{2 \times 2})$$
3. Third Invariant (Total Conic Discriminant): $$I_3 \equiv \Delta \equiv \det(\mathbf{A}) = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = abc + 2fgh - af^2 - bg^2 - ch^2$$

1. Non-Degenerate vs. Degenerate Conics

The total discriminant $\Delta = \det(\mathbf{A})$ establishes the primary topological branch:

• Degenerate Conics ($\Delta = 0$): The curve factors into lines or a single point:
  • $D = ab - h^2 < 0$: Pair of intersecting real straight lines.
  • $D = ab - h^2 = 0$: Pair of parallel or coincident straight lines ($g^2 - ac \ge 0$).
  • $D = ab - h^2 > 0$: A single isolated point (pair of imaginary intersecting lines).
• Non-Degenerate Proper Conics ($\Delta \ne 0$): The curve represents a genuine conic section:
  • Ellipse ($D = ab - h^2 > 0$):
    • Real Ellipse if $\Delta / (a + b) < 0$.
    • Imaginary Ellipse if $\Delta / (a + b) > 0$.
    • Circle if $a = b$ and $h = 0$.
  • Parabola ($D = ab - h^2 = 0$): An open curve extending to infinity with a single axis of symmetry.
  • Hyperbola ($D = ab - h^2 < 0$): An open curve with two separate branches and two real asymptotes.
    • Rectangular (Equilateral) Hyperbola if $a + b = 0$ (asymptotes at right angles).

1. Determining the Center $(\bar{x}, \bar{y})$

A conic is said to be a central conic if it possesses a center of symmetry $(\bar{x}, \bar{y})$ such that any chord passing through the center is bisected by it. This requires $D = ab - h^2 \ne 0$ (holding for all ellipses and hyperbolas). The center is found by equating partial derivatives to zero: $$\frac{\partial F}{\partial x} = 2(ax + hy + g) = 0 \implies ax + hy + g = 0$$ $$\frac{\partial F}{\partial y} = 2(hx + by + f) = 0 \implies hx + by + f = 0$$ Solving by Cramer's rule: $$\bar{x} = \frac{hf - bg}{ab - h^2}, \qquad \bar{y} = \frac{gh - af}{ab - h^2}$$

2. Translating Origin to the Center

Translating the coordinate axes to the center by substituting $x = X + \bar{x}, y = Y + \bar{y}$ eliminates the linear terms ($gX, fY$). The transformed equation becomes: $$a X^2 + 2h XY + b Y^2 + c' = 0$$ where the new constant term $c'$ is given by: $$c' = g\bar{x} + f\bar{y} + c = \frac{\Delta}{ab - h^2} = \frac{\Delta}{D}$$ Notice the immense elegance of this result: the constant term is simply the quotient of the two fundamental invariants $\Delta / D$!

1. Rotational Diagonalization via Eigenvalues

To reduce the central equation $a X^2 + 2h XY + b Y^2 + c' = 0$ to principal axes, we rotate the axes through angle $\theta$ given by: $$\tan 2\theta = \frac{2h}{a - b}$$ In matrix terms, this is equivalent to diagonalizing the symmetric matrix $\mathbf{A}_{2 \times 2} = \begin{pmatrix} a & h \\ h & b \end{pmatrix}$. The eigenvalues $\lambda_1, \lambda_2$ are the roots of the characteristic equation: $$\det(\lambda \mathbf{I} - \mathbf{A}_{2 \times 2}) = 0 \iff \lambda^2 - (a + b)\lambda + (ab - h^2) = 0$$ $$\lambda_1, \lambda_2 = \frac{(a + b) \pm \sqrt{(a - b)^2 + 4h^2}}{2}$$

2. The Standard Canonical Form

Referred to the principal axes $(X', Y')$, the cross term vanishes identically: $$\lambda_1 X'^2 + \lambda_2 Y'^2 + c' = 0 \iff \frac{X'^2}{-c'/\lambda_1} + \frac{Y'^2}{-c'/\lambda_2} = 1$$

• If $\lambda_1, \lambda_2$ have the same sign (and opposite to $c'$), the curve is an ellipse with semi-axes $A = \sqrt{-c'/\lambda_1}, B = \sqrt{-c'/\lambda_2}$.
• If $\lambda_1, \lambda_2$ have opposite signs, the curve is a hyperbola.

1. The Parabola Condition ($ab - h^2 = 0$)

When $ab - h^2 = 0$, the quadratic terms form a perfect square: $$ax^2 + 2hxy + by^2 = (\alpha x + \beta y)^2, \quad \text{where } \alpha = \sqrt{a}, \; \beta = \sqrt{b}, \; \text{and } \alpha\beta = h$$ Because $D = 0$, the conic has no finite center; it is a non-central conic (parabola).

2. The Systematic Reduction Algorithm

To reduce the parabola to canonical form $Y'^2 = 4AX'$:

1. Group the perfect square: $(\alpha x + \beta y)^2 = -2gx - 2fy - c$.
2. Introduce an arbitrary parameter $\lambda$: $$(\alpha x + \beta y + \lambda)^2 = 2(\lambda \alpha - g)x + 2(\lambda \beta - f)y + (\lambda^2 - c)$$
3. Choose $\lambda$ so that the lines $\alpha x + \beta y + \lambda = 0$ (axis of the parabola) and $2(\lambda \alpha - g)x + 2(\lambda \beta - f)y + (\lambda^2 - c) = 0$ (tangent at vertex) are strictly perpendicular: $$\alpha \cdot 2(\lambda \alpha - g) + \beta \cdot 2(\lambda \beta - f) = 0 \implies \lambda(\alpha^2 + \beta^2) = \alpha g + \beta f \implies \lambda = \frac{\alpha g + \beta f}{a + b}$$
4. Divide each side by $\sqrt{\alpha^2 + \beta^2}$ and $\sqrt{4(\lambda\alpha - g)^2 + 4(\lambda\beta - f)^2}$ respectively to obtain normalized perpendicular distances, reducing immediately to the canonical form $Y'^2 = 4AX'$!