Conics and Quadrics - Key Properties

The rich structure of conics emerges from their dual characterization: geometric (via cones) and algebraic (via equations). Understanding their properties requires both perspectives.

DefinitionTangent and Normal

At any non-singular point $P$ on a conic, there exists a unique tangent line. For the conic $F(x,y) = 0$ , the tangent at $(x_0, y_0)$ has equation:

\frac{\partial F}{\partial x}\bigg|_{(x_0,y_0)} (x - x_0) + \frac{\partial F}{\partial y}\bigg|_{(x_0,y_0)} (y - y_0) = 0

The normal is perpendicular to the tangent and passes through $P$ .

Reflective properties distinguish different conics:

Ellipse: Light from one focus reflects through the other focus
Parabola: Light from the focus reflects parallel to the axis
Hyperbola: Light aimed at one focus reflects toward the other focus

These optical properties have practical applications in telescope design, satellite dishes, and acoustic engineering.

DefinitionPole and Polar

Given a conic $C$ and a point $P$ (possibly not on $C$ ), the polar of $P$ is the locus of harmonic conjugates of $P$ with respect to pairs of points where lines through $P$ intersect $C$ .

Conversely, given a line $\ell$ , its pole is the point whose polar is $\ell$ . This establishes a duality between points and lines relative to the conic.

The relationship between eccentricity and conic type reflects the balance between radial and angular properties. Eccentricity $e = 0$ gives a circle (perfect symmetry), $0 < e < 1$ gives ellipses (bounded curves), $e = 1$ gives parabolas (critical case), and $e > 1$ gives hyperbolas (unbounded with asymptotes).

ExampleIntersection Properties

Bézout's Theorem applied to conics: Two conics in the projective plane intersect in exactly 4 points (counting multiplicity and complex intersections). This follows from their degree being 2, giving $2 \times 2 = 4$ intersection points.

Special cases include:

Tangent conics share a tangent line at one point (multiplicity 2)
Concentric circles intersect at 2 complex points at infinity

Confocal conics (conics sharing the same foci) form an orthogonal family: every point in the plane (except foci) lies on exactly one ellipse and one hyperbola from a confocal family, and these curves are perpendicular at the intersection.

Remark

Matrix representation unifies conic equations. A conic $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ corresponds to the quadratic form:

\mathbf{x}^T Q \mathbf{x} = 0 \quad \text{where } \mathbf{x} = \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}, \quad Q = \begin{pmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{pmatrix}

The eigenvalues of $Q$ determine the conic's classification and orientation.

Pascal's and Brianchon's theorems provide powerful incidence properties for points on conics, unifying many special cases and revealing the deep projective structure underlying conic geometry.