Conic Sections · ConicLab

Slicing a cone

Slice a double cone with a plane and the curve depends on the tilt: perpendicular to the axis gives a circle, a slight tilt an ellipse, parallel to a generator a parabola, and cutting both nappes a hyperbola.

If the plane makes angle β with the axis and the cone's half-angle is α, the eccentricity is e = cos β / cos α: β > α ellipse, β = α parabola, β < α hyperbola.

Dandelin spheres

Place two spheres inside the cone, each tangent to the cone and to the cutting plane. Where they touch the plane, F₁ and F₂, are exactly the foci of the curve.

Tangent segments from an outside point to a sphere are equal. So PF₁ equals PQ₁, the distance from P along its generator to the upper tangent circle; likewise PF₂ = PQ₂.

Hence PF₁ + PF₂ = Q₁Q₂, the generator length between the two circles, independent of P: the definition of an ellipse.

Ellipse

An ellipse is the set of points whose distances to two fixed points F₁, F₂ add up to a constant 2a (2a > |F₁F₂|).

Pin a string at the foci, pull it taut with a pencil and go round: that draws an ellipse.

Standard form x²/a² + y²/b² = 1 (a > b > 0) with a² = b² + c². The closer e = c/a is to 1, the flatter the ellipse.

Hyperbola

A hyperbola is the set of points whose distances to two fixed points F₁, F₂ differ by a constant 2a (0 < 2a < |F₁F₂|) in absolute value.

Standard form x²/a² − y²/b² = 1 with c² = a² + b². It has two asymptotes y = ±(b/a)x that the branches hug farther out.

When a = b it is a rectangular hyperbola: the asymptotes are perpendicular and e = √2.

Parabola

A parabola is the set of points equidistant from a fixed point F and a fixed line l (F not on l). F is the focus, l the directrix.

Standard form y² = 2px (p > 0): focus (p/2, 0), directrix x = −p/2; p is the focus-to-directrix distance.

The chord through the focus perpendicular to the axis is the latus rectum, of length 2p.

Eccentricity

All three curves share one definition: the points whose distance to a fixed point F divided by their distance to a fixed line l is a constant e.

0 < e < 1 gives an ellipse, e = 1 a parabola, e > 1 a hyperbola; the closer e is to 0, the rounder the ellipse.

Drag the e slider and watch the curve morph from circle to ellipse, parabola and hyperbola.

Reflective property

Light from one focus of an ellipse reflects into the other focus.

Light from the focus of a parabola reflects parallel to the axis, the principle behind searchlights and satellite dishes.

Light from one focus of a hyperbola reflects so that its backward extension passes through the other focus.

Drag the light off the focus and the reflections stop converging.

Lines and conics

Substitute y = kx + m into the curve and eliminate y to get an equation in x. When the x² coefficient is non-zero, the discriminant Δ decides: Δ > 0 two intersections, Δ = 0 tangent, Δ < 0 none.

Chord length |AB| = √(1 + k²)·|x₁ − x₂|. For an ellipse with chord midpoint M, kAB · kOM = −b²/a².