Conic sections

The formula

The focus-directrix definition, which is the whole family in one line:

distance to focus  =  e · distance to directrix

Reading: pick a point (focus), a line (directrix), and a ratio e. Every point holding that ratio traces a conic, and e alone decides which kind.

The classification:

e = 0        circle
0 < e < 1    ellipse        (closed, bound)
e = 1        parabola       (open, the boundary case)
e > 1        hyperbola      (open, two branches)

Reading: eccentricity is a dial, not a category. Turn it up from zero and the circle stretches, breaks open at exactly e = 1, and splits into a hyperbola beyond.

The polar form, focus at the origin — the working equation of orbital mechanics:

r(θ) = p / (1 + e·cosθ)          p = a·(1 − e²)  for an ellipse

Reading: r is distance from the focus, θ is angle from closest approach, p (semi-latus rectum) sets the size. This single equation is every two-body trajectory ever flown.

The Cartesian standard forms, axes aligned and centered:

ellipse:    x²/a² + y²/b² = 1        foci at (±c, 0),  c² = a² − b²,  e = c/a
parabola:   y² = 4·f·x               focus at (f, 0), directrix x = −f
hyperbola:  x²/a² − y²/b² = 1        foci at (±c, 0),  c² = a² + b²,  e = c/a

Reading: one sign flip separates the ellipse from the hyperbola; the parabola is what lives on the flip itself.

The test for a conic in the wild — any second-degree curve is one:

A·x² + B·xy + C·y² + D·x + E·y + F = 0
B² − 4AC < 0   ellipse        B² − 4AC = 0   parabola        B² − 4AC > 0   hyperbola

Reading: fit a general quadratic to your data and the discriminant names the curve before you ever plot it (degenerate cases — a point, crossed lines — excepted).

Where you meet it

At the mission design review. Every trajectory slide is this one equation wearing different eccentricities. A geostationary transfer orbit from a 185 km parking altitude to 35,786 km runs e ≈ 0.73; an interplanetary departure is a hyperbola with respect to Earth; the boundary between them is the parabola, which is another name for exactly escape velocity. The reviewer asking "what's the eccentricity?" is asking which member of the family you're on.
On the antenna range. A reflector dish is a paraboloid because a parabola is the only curve that turns a spherical wave from its focus into a plane wave — same property, run backward, that focuses an incoming beam onto the feed. Bench check: lay a straightedge across a dish of diameter D, measure the depth d at center, and the focal length is f = D²/(16·d). A 1 m dish 15 cm deep puts the feed 41.7 cm out. If the horn isn't there, gain isn't either.
At the CMM, checking a cam or a mirror. Cam profiles, optical surfaces, and pressure-vessel heads get inspected by fitting the general quadratic to probe points. The discriminant then tells you whether the shop actually cut the ellipse on the drawing or something subtly parabolic — a distinction the eye cannot make and the follower dynamics can.
In the link-budget review. The Fresnel clearance zone around a microwave path is an ellipsoid with a focus at each antenna — the constant-total-path-length (constant-sum) definition of the ellipse, doing propagation work: every point on the first zone's boundary makes the antenna-to-antenna path exactly a half-wavelength longer than line-of-sight. A ridge that grazes the ellipse's belly costs you dB the free-space equation never predicted.

How it works

The cone picture earns its keep once you see why the foci matter. Tilt a cutting plane through a cone: a shallow cut closes into an ellipse, a cut parallel to the cone's slope just fails to close — parabola — and a steeper cut meets both nappes of the double cone, giving the hyperbola its two branches. In 1822 a Belgian military engineer named Germinal Dandelin showed the punchline: inscribe spheres in the cone tangent to the cutting plane, and the points where they touch the plane are the foci ^[7]^[8]. The slicing definition and the focus definition are the same fact.

The reflection property is the family's working asset. Rays leaving one focus of an ellipse arrive at the other; rays leaving a parabola's focus leave parallel to the axis; rays aimed at one focus of a hyperbola bounce toward the other ^[1]. A Cassegrain antenna is this property used twice — parabolic primary, hyperbolic secondary sharing a focus with it — which is why the feed can sit conveniently behind the vertex instead of hanging out at the prime focus in the wind.

In orbit work, eccentricity is energy in disguise. Bound (negative-energy) trajectories are ellipses, unbound ones hyperbolas, and e = 1 is the knife edge between them ^[6]. Which is the first gotcha: a true parabolic orbit is a measure-zero idealization. Real hardware is always slightly bound or slightly unbound, and near e = 1 small velocity errors swing the predicted trajectory wildly — the polar equation's denominator is heading toward zero at wide angles. The same knife edge bites curve-fitting: the discriminant of a fitted quadratic sits near zero for anything parabola-like, so noise flips the classification between ellipse and hyperbola from one dataset to the next. Don't let a fit routine tell you an orbit's family; let the energy do it.

Second gotcha: eccentricity is not visual flattening. Earth's orbit has e = 0.0167 and the ratio b/a = √(1 − e²) = 0.99986 — drawn to scale it is a circle to any eye. What the eccentricity actually moves is the Sun, sitting c = a·e off center, which is why perihelion distance varies by 3.3% while the orbit's shape barely deviates from round. Textbook diagrams with dramatically squashed ellipses have miscalibrated more intuitions than any equation.

Third: the projectile "parabola" is a flat-Earth approximation. In uniform gravity the trajectory is parabolic; in the real inverse-square field it is a slice of a very long ellipse with one focus at Earth's center. For a mortar round the difference is nothing; for anything ballistic at range, the flat-Earth parabola is the error term the trajectory shop exists to remove.

Last: in the hyperbola, b is not a distance you can point to on the curve. It sets the asymptote slope ±b/a — the direction a flyby trajectory straightens out toward — and people who carry ellipse intuition across the sign flip routinely misread it.

History

The family is old enough that its discovery problem was a religious one. Menaechmus, a Greek geometer working around 350 BC (roughly 380–320 BC), found the curves while attacking the Delian problem — doubling the volume of a cubical altar — and solved it by intersecting a parabola with a hyperbola, treating both as sections of a cone ^[1]^[2]. About a century and a half later Apollonius of Perga (c. 262–190 BC) wrote the eight-book Conics, proved that any plane cutting a double cone yields one of the family, and gave the curves the names we still use: ellipse, parabola, hyperbola ^[1]^[3]. Books five through seven, on normals and curvature, were original enough that only a handful of mathematicians in the following eighteen centuries added anything.

The curves stayed pure geometry until an astronomer needed them. Johannes Kepler coined the term "focus" — Latin for hearth or burning-point, from work on burning mirrors — in his 1604 optics treatise ^[1]^[9]. Five years later, after his self-described "war with Mars" against Tycho Brahe's observation logs, he published Astronomia Nova (1609) with the claim that broke two thousand years of circular-orbit doctrine: Mars moves on an ellipse with the Sun at one focus ^[4]^[5]. Newton's Principia (1687) supplied the why, deriving Kepler's laws from inverse-square gravitation — under which every free trajectory is a conic, with eccentricity set by the body's energy ^[5]^[6]. Dandelin's spheres closed the loop in 1822, tying the ancient slicing definition to the foci the astronomers had been navigating by ^[7]^[8]. Apollonius wrote the flight manual for spacecraft eighteen centuries before anyone had a use for it.

Related tools

/tools/orbital-velocity-period — circular case, e = 0, of the polar orbit equation
/tools/hohmann-transfer — a transfer orbit is half an ellipse; the GTO example above is this calculator's territory
/tools/escape-velocity — the speed at which the conic becomes a parabola, exactly e = 1
/tools/tsiolkovsky-delta-v — the propellant cost of moving between members of the family
/tools/antenna-gain-beamwidth — the payoff of the parabola's reflection property
/tools/fresnel-zone — the ellipsoid with a focus at each end of the link
/tools/projectile-range — the flat-Earth parabola, with the caveat noted above

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →