Conics and Quadrics - Core Definitions

Conic sections—curves obtained by intersecting a cone with a plane—have fascinated mathematicians since antiquity. These curves exhibit remarkable geometric and algebraic properties that make them fundamental in pure and applied mathematics.

DefinitionConic Section

A conic section (or simply conic) is a curve obtained by intersecting a double circular cone with a plane. Depending on the angle of intersection:

Circle: plane perpendicular to the cone's axis
Ellipse: plane cuts through one nappe at an angle
Parabola: plane parallel to a generating line
Hyperbola: plane cuts through both nappes

Algebraically, conics are zero sets of degree-2 polynomials in two variables.

In Cartesian coordinates, the general equation of a conic is:

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

The discriminant $\Delta = B^2 - 4AC$ determines the type:

$\Delta < 0$ : ellipse (or circle if $A = C$ and $B = 0$ )
$\Delta = 0$ : parabola
$\Delta > 0$ : hyperbola

DefinitionFocus-Directrix Property

A conic can be defined as the locus of points $P$ such that the ratio of distances from a fixed point $F$ (focus) to a fixed line $\ell$ (directrix) is constant:

\frac{d(P,F)}{d(P,\ell)} = e

The constant $e$ is the eccentricity:

$e < 1$ : ellipse
$e = 1$ : parabola
$e > 1$ : hyperbola

The ancient Greeks (Apollonius, circa 200 BCE) studied conics extensively, deriving their properties synthetically. They discovered that all conics arise from the same cone, just with different cutting planes. This unified treatment presaged modern abstract approaches.

ExampleStandard Forms

In standard position, conics have simple equations:

Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (foci at $(\pm c, 0)$ where $c^2 = a^2 - b^2$ )

Parabola: $y^2 = 4px$ (focus at $(p, 0)$ , directrix $x = -p$ )

Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (foci at $(\pm c, 0)$ where $c^2 = a^2 + b^2$ )

In projective geometry, all non-degenerate conics are equivalent—there's only one conic up to projective transformation. This unifies the classification: the distinction between ellipse, parabola, and hyperbola depends on how the conic intersects the line at infinity.

Remark

Conics appear throughout science and engineering:

Planetary orbits: Kepler's first law states planets move in elliptical orbits
Reflective properties: Parabolic mirrors focus parallel light rays at a point
Optics: Elliptical reflectors have two foci; light from one reflects through the other
Architecture: Parabolic arches and hyperbolic cooling towers exploit structural properties

Quadric surfaces generalize conics to three dimensions. A quadric is the zero set of a degree-2 polynomial in three variables:

Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0

Standard forms include ellipsoids, hyperboloids, paraboloids, and cylinders.