Conic Sections

The four curves

Slice a double cone with a plane. Depending on the angle of the slice:

• Plane perpendicular to cone's axis → circle.
• Plane tilted slightly (less than the cone's slant angle) → ellipse (closed oval).
• Plane parallel to the cone's slant edge → parabola (open curve, single branch).
• Plane tilted past the slant angle, cutting both cones → hyperbola (open curve, two branches).

Discovered by Apollonius of Perga around 200 BCE in his treatise Conics — the foundational work on the subject. He named them: ellipse ("falling short"), parabola ("alongside"), hyperbola ("exceeding").

Standard equations

Conic	Standard form	Eccentricity e	Geometry
Circle	x² + y² = r²	0	Center origin, radius r
Ellipse	x²/a² + y²/b² = 1	√(1 − b²/a²) (a>b)	Semi-axes a, b; foci at (±c, 0), c² = a² − b²
Parabola	y² = 4px	1 (exactly)	Focus at (p, 0), directrix x = −p
Hyperbola	x²/a² − y²/b² = 1	√(1 + b²/a²)	Vertices (±a, 0); asymptotes y = ±(b/a)x

Unified algebraic form

Every conic is the zero set of a quadratic in two variables:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The discriminant B² − 4AC classifies the conic:

• B² − 4AC < 0 — ellipse (or circle when A = C and B = 0).
• B² − 4AC = 0 — parabola.
• B² − 4AC > 0 — hyperbola.

Degenerate cases (a point, a line, two lines) occur when the equation factors or describes degenerate sections of the cone (passing through the apex).

Focus-directrix unified definition

Pick a point F (the focus) and a line ℓ (the directrix) not containing F. Pick a positive number e (the eccentricity).

The locus of points P such that |PF| / d(P, ℓ) = e is a conic:

• e = 0 — circle (degenerate; the focus is the center).
• 0 < e < 1 — ellipse.
• e = 1 — parabola.
• e > 1 — hyperbola.

This unifies all conics into a single parameter family.

Parabolic reflectors

The parabola has a special reflective property — any ray of light (or sound, or radio wave) traveling parallel to the parabola's axis reflects off the parabola and passes through the focus.

Conversely, any ray emitted from the focus reflects off the parabola and travels parallel to the axis. This makes parabolas perfect for:

• Satellite dishes — incoming parallel signals focus to a single receiver point.
• Car headlights — bulb at focus, beam emerges parallel.
• Solar concentrators — focus sunlight onto a small target for heating.
• Parabolic microphones — capture distant sounds by focusing them.

Kepler orbits — conics in physics

Kepler's first law (1609) — planets orbit the Sun in ellipses with the Sun at one focus.

Newton (1687) derived this from the inverse-square law of gravity — under F = −GMm/r², bound orbits are ellipses (or circles, the e=0 limit), and unbound trajectories are parabolas (exactly escape velocity) or hyperbolas (excess velocity).

Object	Trajectory	Eccentricity
Earth	Ellipse	0.0167 (very nearly circular)
Mars	Ellipse	0.0934
Halley's Comet	Ellipse	0.967 (highly elongated)
Voyager 1 (after Saturn)	Hyperbola	3.7 (interstellar escape)
'Oumuamua	Hyperbola	1.20 (came from interstellar space)

JavaScript — drawing conics

// Parametric — generate points on each conic
function circlePoints(r, n = 100) {
  return Array.from({length: n + 1}, (_, i) => {
    const t = (i / n) * 2 * Math.PI;
    return [r * Math.cos(t), r * Math.sin(t)];
  });
}

function ellipsePoints(a, b, n = 100) {
  return Array.from({length: n + 1}, (_, i) => {
    const t = (i / n) * 2 * Math.PI;
    return [a * Math.cos(t), b * Math.sin(t)];
  });
}

function parabolaPoints(p, tMin = -3, tMax = 3, n = 100) {
  return Array.from({length: n + 1}, (_, i) => {
    const t = tMin + (i / n) * (tMax - tMin);
    return [2 * p * t, p * t * t];
  });
}

function hyperbolaPoints(a, b, branch = 1, tMin = -2, tMax = 2, n = 100) {
  return Array.from({length: n + 1}, (_, i) => {
    const t = tMin + (i / n) * (tMax - tMin);
    return [branch * a * Math.cosh(t), b * Math.sinh(t)];
  });
}

// Classify a conic from general form: Ax² + Bxy + Cy² + Dx + Ey + F = 0
function classifyConic(A, B, C, D, E, F) {
  const disc = B * B - 4 * A * C;
  if (Math.abs(disc) < 1e-10) return 'parabola';
  if (disc < 0) return A === C && B === 0 ? 'circle' : 'ellipse';
  return 'hyperbola';
}

console.log(classifyConic(1, 0, 1, 0, 0, -25));  // x² + y² = 25 → 'circle'
console.log(classifyConic(4, 0, 9, 0, 0, -36));  // 4x² + 9y² = 36 → 'ellipse'
console.log(classifyConic(0, 0, 1, -4, 0, 0));   // y² = 4x → 'parabola'
console.log(classifyConic(1, 0, -1, 0, 0, -1));  // x² - y² = 1 → 'hyperbola'

Where conics show up

• Astronomy. Kepler orbits, comet trajectories, planetary mechanics. Gravitational two-body problem produces conic sections.
• Engineering optics. Parabolic reflectors, elliptical mirrors (whispering galleries), hyperbolic lenses.
• Architecture. Arches and domes use parabolic and elliptical curves; the Gateway Arch in St. Louis is a catenary, not a parabola, but visually similar.
• Navigation — GPS and LORAN. Hyperbolic positioning — distance differences from receivers form hyperbolas.
• Projectile motion. Idealized projectiles (no air resistance) follow parabolas. Real projectiles approximate parabolas at low speeds.
• Computer graphics — Bézier and NURBS. Conic sections appear as special cases of rational Bézier curves; arcs and circles are exact when implemented carefully.
• Algebraic geometry. Foundational examples — conics are the simplest non-trivial algebraic curves; their study generalizes to elliptic curves, cubic curves, etc.

Common mistakes

• Confusing ellipse with oval. "Oval" is informal; "ellipse" is precise (the locus of points whose distances to two foci sum to a constant). Eggs are oval, not elliptical.
• Forgetting the cross term Bxy. A conic rotated off the standard axes has B ≠ 0. Diagonalize (rotate coordinates) to remove the cross term, then classify.
• Mistaking hyperbola asymptotes for the curve. The asymptotes y = ±(b/a)x are not part of the hyperbola; the curve approaches but never touches them.
• Confusing eccentricity with elongation. Eccentricity 0 means circular; eccentricity approaching 1 means highly elongated ellipse, but eccentricity 1 itself is parabola, not the most-elongated ellipse.
• Treating the focus as the center. The center of an ellipse is the midpoint of the two foci, not at a focus. The Sun is at a focus, not at Earth's orbital center.
• Assuming "parabolic" trajectory ignores air resistance. Real projectile motion with drag is not parabolic; it's a more complex curve. The parabola approximation works only at low speeds with small drag.