Conic Sections: One Cone, Four Curves, One Equation

One cone, one plane, four curves

Slice a double cone — two nappes joined tip to tip — with a flat plane and the cross-section is always one of four shapes: a circle when the plane is perpendicular to the axis, an ellipse when it is tilted but still crosses only one nappe, a parabola when it tilts to become exactly parallel to a line on the cone's surface, and a hyperbola when it tilts further still and slices through both nappes at once. The Greek geometer Apollonius of Perga worked out this whole taxonomy around 200 BCE, purely from the solid geometry of the cone, over eighteen centuries before Kepler discovered that these same four curves describe every possible orbit under an inverse-square force.

Eccentricity: one number, four curves

The modern, algebraically cleaner way to unify the four curves is eccentricity, e — the ratio of a point's distance from a fixed focus to its distance from a fixed line (the directrix). e = 0 is a circle, 0 < e < 1 is an ellipse, e = 1 is a parabola, and e > 1 is a hyperbola. In terms of the slicing cone, if α is the cone's own half-angle and θ is the tilt of the cutting plane from the axis, then:

e = sin(theta) / sin(alpha)

theta = 90 deg (plane perpendicular to axis)      -> e = 1/sin(alpha) ... wait, perpendicular plane: theta=alpha  -> e=1 -> circle at theta=alpha?
(convention varies; the working rule used in the simulation is e = cos(theta)/cos(alpha), theta measured from the base)
theta < alpha   -> ellipse   (0 <= e < 1)
theta == alpha   -> parabola  (e == 1)
theta > alpha   -> hyperbola (e > 1)

Every conic also has a clean polar equation centred on a focus, r = ℓ / (1 + e·cos φ), where ℓ is the semi-latus rectum. This single formula is the one Newton and Kepler actually used for orbital mechanics — plug in e for any planet, comet or spacecraft trajectory and the same equation produces the ellipse of a bound planet, the parabola of an object at exactly escape velocity, or the hyperbola of an interstellar visitor like ʻOumuamua that passes through once and never returns.

Why an ellipse has two foci and a taut string

The classic pins-and-string construction of an ellipse — loop a string around two pins and trace with a pencil that keeps it taut — works because an ellipse is defined as the set of points where the sum of distances to two fixed foci is constant. Dandelin's spheres, discovered in 1822, give the beautiful proof of why this matches the cone-slicing definition: inscribe a sphere in each nappe of the cone, tangent to the cutting plane at one point each. Those two tangent points turn out to be exactly the ellipse's two foci, and simple tangent-line-length arguments show the sum of distances to them is constant along the whole curve.

A hyperbola is the same idea with a minus sign

A hyperbola swaps the sum for a difference: the set of points where the difference of distances to two foci is a constant. It has two separate branches because that difference can be positive or negative, and its two asymptotes are the lines the curve approaches but never touches as it runs to infinity — for the standard form x²/a² − y²/b² = 1 the asymptotes are simply y = ±(b/a)x. A parabola sits at the exact threshold between the two: defined by equal distance to a single focus and a single directrix line, it is the unique curve with only one focus, which is also why it is the only one of the four that is not a closed or two-branched curve but a single open arc reaching infinity in one consistent direction.

Where the four curves show up in the real world

Ellipses are every closed orbit under gravity, the shape of a whispering gallery, and the cross-section of a cylinder cut on the diagonal. Parabolas are the trajectory of anything thrown in a uniform gravitational field with no air resistance, and the shape used in satellite dishes and car headlight reflectors because a parabola focuses all rays parallel to its axis onto one point (or, run in reverse, emits a perfectly collimated beam from a source at that focus). Hyperbolas describe unbound orbits — anything moving faster than escape velocity past a gravitating body — and are also the shape LORAN and modern multilateration systems use, since the locus of points with a constant time-difference between two fixed radio transmitters is, by definition, one branch of a hyperbola.

Frequently asked questions

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