Conic Sections

Q: Why does a satellite dish work?

Because of the parabola's reflective property: any incoming ray parallel to the axis reflects to the focus. Signals from a distant satellite arrive as essentially parallel rays; the dish focuses them all on the receiving antenna placed at the focus. Same principle is used in radio telescopes, headlight reflectors, and solar cookers.

Q: How are conic sections related to planetary orbits?

Newton proved that any orbit under inverse-square gravity is a conic section. Bound orbits (with enough kinetic energy to escape, etc.) are ellipses (or circles as a special case). Marginally bound orbits are parabolas. Unbound trajectories — interstellar visitors, gravity-assist flybys — are hyperbolas. Kepler's first law (planets orbit in ellipses with the Sun at a focus) is the bound case.

Q: Did the Greeks really study conics 2000 years ago?

Yes. Apollonius of Perga wrote a treatise titled 'Conics' around 200 BCE that systematically developed the theory. The Greeks worked purely geometrically — coordinates and algebra came much later (Descartes, 17th century). Apollonius's results are still essentially correct and form the basis of modern conic-section theory.

Conic sections are the curves obtained by slicing a double cone with a plane. Depending on the angle of the cut, you get one of four shapes: a circle, an ellipse, a parabola, or a hyperbola. Studied by the Greeks more than 2,000 years ago, these curves describe the orbits of planets, the path of a projectile, the design of telescope mirrors, and the focusing properties of satellite dishes and car headlights. The unifying algebraic form is the second-degree equation in two variables.

Conic sections — a double cone intersected at four angles producing circle, ellipse, parabola, and hyperbola with their standard equations. — The four conic sections produced by slicing a cone at different angles: circle, ellipse, parabola, hyperbola.

Four Curves From One Cone

Imagine an infinite double cone with its vertex at the origin. Slice it with a plane:

Circle. Plane perpendicular to the cone’s axis. Equation: $ x^2 + y^2 = r^2 $.
Ellipse. Plane tilted but not parallel to the side of the cone. Equation: $ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 $.
Parabola. Plane parallel to one side of the cone. Equation: $ y = ax^2 + bx + c $ (or $ x = ay^2 + by + c $).
Hyperbola. Plane parallel to the axis (slicing both halves of the cone). Equation: $ \dfrac{x^2}{a^2} – \dfrac{y^2}{b^2} = 1 $.

The General Equation

All four conics are described by a single second-degree polynomial in $ x $ and $ y $:

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

The discriminant $ B^2 – 4AC $ classifies the curve:

$ B^2 – 4AC < 0 $: ellipse (circle if $ B = 0 $ and $ A = C $).
$ B^2 – 4AC = 0 $: parabola.
$ B^2 – 4AC > 0 $: hyperbola.

The Focus-Directrix Definition

Every conic can be defined as the locus of points whose distance to a fixed focus $ F $ and a fixed directrix line $ d $ have a constant ratio $ e $ — the eccentricity:

$$ e = \frac{\text{distance to focus}}{\text{distance to directrix}} $$

$ e = 0 $: circle.
$ 0 < e < 1 $: ellipse.
$ e = 1 $: parabola.
$ e > 1 $: hyperbola.

Reflective Properties

Parabola. Any ray parallel to the axis reflects off the parabola and passes through the focus. This is why parabolic dishes are used for satellite reception, radio telescopes, headlight reflectors, and solar concentrators — they collect parallel incoming energy at a single point.
Ellipse. Any ray emitted from one focus reflects off the ellipse and passes through the other focus. This is the principle behind ‘whispering galleries’ (Statuary Hall in the US Capitol, St. Paul’s Cathedral) and medical lithotripsy machines that focus shock waves to break up kidney stones.
Hyperbola. A ray aimed at one focus reflects off the hyperbola as if it had come from the other focus. Used in Cassegrain telescopes and certain GPS positioning calculations.

Orbits and Kepler

Kepler’s first law states that planets orbit the Sun in ellipses with the Sun at one focus. Comets typically follow elongated ellipses or, if they’re unbound, parabolic or hyperbolic paths. Newton derived this directly from his inverse-square law of gravity: the only orbits possible under $ 1/r^2 $ attraction are conic sections. Eccentricity tells you which: planets and most asteroids $ e < 1 $, comets often $ e $ near 1, interstellar visitors $ e > 1 $.

Applications

Astronomy. Every gravitationally bound orbit is an ellipse; every gravity-assist trajectory through a planet’s gravity well is a hyperbola.
Engineering. Parabolic arches in bridges (Sydney Harbour, classic stone arches) and suspension bridge cables (catenary, very close to parabolic when loaded uniformly).
Optics. Reflective telescopes (Newtonian, Cassegrain) use parabolic primary mirrors. Radar antennas and satellite dishes are parabolic for the same focusing reason.
Acoustics. Whispering galleries shaped as elliptical domes: sound from one focus carries with surprising clarity to the other focus.
Computer graphics and CAD. Bézier and NURBS curves used for vector graphics and 3D modeling are direct generalizations of conics.

Frequently Asked Questions

What are the four conic sections?

Circle, ellipse, parabola, and hyperbola. They’re obtained by slicing a double cone with a plane at different angles: perpendicular to the axis gives a circle, slightly tilted gives an ellipse, parallel to the cone’s side gives a parabola, and parallel to the axis gives a hyperbola.

What’s the general equation of a conic section?

Ax² + Bxy + Cy² + Dx + Ey + F = 0. The type is determined by the discriminant B² − 4AC: negative gives an ellipse (circle if also B = 0 and A = C), zero gives a parabola, positive gives a hyperbola.

What is eccentricity?

A number that describes how ‘stretched’ a conic is. It’s the ratio of the distance from any point on the curve to a focus over the distance to a directrix. Circle: e = 0. Ellipse: 0 < e 1. Planets have very low eccentricity (Earth’s is 0.017); Halley’s comet has e ≈ 0.967.

Why does a satellite dish work?

Because of the parabola’s reflective property: any incoming ray parallel to the axis reflects to the focus. Signals from a distant satellite arrive as essentially parallel rays; the dish focuses them all on the receiving antenna placed at the focus. Same principle is used in radio telescopes, headlight reflectors, and solar cookers.

How are conic sections related to planetary orbits?

Newton proved that any orbit under inverse-square gravity is a conic section. Bound orbits (with enough kinetic energy to escape, etc.) are ellipses (or circles as a special case). Marginally bound orbits are parabolas. Unbound trajectories — interstellar visitors, gravity-assist flybys — are hyperbolas. Kepler’s first law (planets orbit in ellipses with the Sun at a focus) is the bound case.

Did the Greeks really study conics 2000 years ago?

Yes. Apollonius of Perga wrote a treatise titled ‘Conics’ around 200 BCE that systematically developed the theory. The Greeks worked purely geometrically — coordinates and algebra came much later (Descartes, 17th century). Apollonius’s results are still essentially correct and form the basis of modern conic-section theory.