Ellipse

The formula

The standard form, centered at the origin with the long axis on x:

x²/a² + y²/b² = 1        (a = semi-major axis, b = semi-minor axis, a ≥ b)

Reading: a circle's x² + y² = r² with two different radii — every point is a circle point scaled by a horizontally and b vertically.

The foci and eccentricity:

c² = a² − b²        e = c/a        (0 ≤ e < 1)

Reading: the two foci sit on the major axis, a distance c either side of center; e is that offset as a fraction of the semi-major axis. e = 0 is a circle. e → 1 flattens the ellipse onto its own major axis.

The defining property, which is also the shop-floor construction:

r₁ + r₂ = 2a        (distances from any point on the curve to the two foci)

Reading: pin a string of length 2a at the two foci, pull it taut with a pencil, and trace. Layout crews still loft elliptical arches this way.

The polar form measured from one focus — the one every orbit calculation uses:

r(θ) = a·(1 − e²) / (1 + e·cosθ)

Reading: θ = 0 gives closest approach r = a(1 − e) (perigee), θ = π gives the far point a(1 + e) (apogee). This is Kepler's first law written as one line.

Area is clean; perimeter is not:

A = π·a·b
C = 4a·E(e)        (complete elliptic integral of the second kind — no closed form)
C ≈ π·(a + b)·(1 + 3h/(10 + √(4 − 3h))),   h = (a − b)²/(a + b)²     (Ramanujan)

Reading: the area formula is exact and obvious once you see the ellipse as a scaled circle. The perimeter integral is why "elliptic integrals" have their name ^[2]. Ramanujan's approximation is good to better than one part in 10⁹ at a 2:1 aspect ratio and about one part in 10⁵ even at 10:1 — more accuracy than any tape measure you own.

Where you meet it

At the rotor test cell, staring at an orbit plot. Two proximity probes mounted 90° apart watch a spinning shaft's centerline, and the X–Y trace on the screen is an ellipse. A healthy machine on anisotropic bearings whirls in a modest ellipse; a flattening orbit points at excessive preload or misalignment, and a growing one at instability. The ellipse's size, flatness, and tilt are the diagnosis.
In the orbit review board slides. Every closed orbit is an ellipse with the central body at one focus, not the center. Perigee is a(1 − e), apogee is a(1 + e). A Molniya-type orbit with roughly 600 km perigee and 39,750 km apogee altitude has e ≈ 0.74 — the spacecraft loiters near apogee and sprints through perigee, which is the whole point of the design.
On the antenna range. Real "circular" polarization is elliptical. The tip of the E-field vector traces an ellipse once per RF cycle, and axial ratio — major axis over minor axis, quoted in dB — is the spec. A 2:1 polarization ellipse is a 6.02 dB axial ratio, and it costs you link margin against a truly circular receive antenna.
Inside an oval-gear flow meter. Two meshed elliptical gears trap and pass fixed pockets of fluid; count rotations, know the volume. Elliptical gears also show up wherever a designer wants a nonuniform output speed from a uniform input.

How it works

The two-focus picture is the one that earns its keep. Everything measured from center is symmetric and dull; everything measured from a focus is where the physics lives. A ray leaving one focus reflects off the ellipse straight through the other focus — that is why whispering galleries work and why elliptical reflectors focus a lamp arc onto a fiber tip.

The mistake people make with eccentricity: assuming small e means nothing interesting is happening. The squash is second order in e — b = a·√(1 − e²) ≈ a·(1 − e²/2) — but the focus offset is first order, c = a·e. Earth's orbit has e ≈ 0.0167. Drawn to scale it is a circle: b falls short of a by 0.014%, less than a line width. But the Sun sits about 2.5 million km off center, so the Sun–Earth distance swings from roughly 147.1 to 152.1 million km over the year — a 3.3% range from a "nearly circular" orbit. Range, timing, and thermal loads all ride on a·e, not on the shape.

Second standing mistake: the perimeter. There is no closed form, full stop — π·(a + b) reads plausible and is 2.7% short at 2:1, 15% short at 10:1. Belt lengths, seal grooves, and elliptical duct linings sized that way come up short. Use 4a·E(e) from any math library, or Ramanujan's formula above when you need one line in a spreadsheet.

Two more traps worth naming. Geodesy uses flattening f = (a − b)/a while orbit work uses eccentricity, and they are not the same number: the WGS-84 Earth ellipsoid has f = 1/298.257 but e ≈ 0.0818 (via e² = 2f − f²). Code that feeds one where the other belongs will run fine and be wrong by miles. And any circle viewed off-axis projects to an ellipse — which means a perfectly circular shaft whirl seen by probes that aren't truly orthogonal, or a round hole photographed at an angle, reads elliptical. Before you diagnose the machine, check the measurement geometry.

Validity limits are simple: all of the above is the plane ellipse. Orbits are ellipses only in the two-body, point-mass idealization — J2 oblateness, drag, and third bodies make real orbits precessing near-ellipses, which is why "osculating elements" exist as a term.

History

The curve is old and the applications kept re-electing it. Apollonius of Perga, working around 200 BC, wrote the eight-book Conics that treated the ellipse as a slice through a cone and gave it its name — Greek for "deficit," because in his construction the ordinate's square falls short of a reference area ^[3]^[4]. The definitions in that book are essentially the ones still taught.

For eighteen centuries the ellipse stayed geometry. Then Johannes Kepler, grinding through Tycho Brahe's Mars observations — nearly a thousand surviving folio sheets of arithmetic he called his "war with Mars" — published Astronomia Nova in 1609 and announced that planets move in ellipses with the Sun at one focus ^[5]^[6]. A pure-geometry object from antiquity turned out to be the shape of the solar system, and every trajectory analyst since has worked in Apollonius's curve with Kepler's focus.

The perimeter resisted everyone. The integral defied closed form and ultimately spawned the theory of elliptic integrals. Srinivasa Ramanujan, in his 1914 paper Modular Equations and Approximations to π, stated the two approximations engineers still use, the better of which is accurate beyond any practical measurement — offered, in his style, without proof ^[7]^[8].

Related tools

/tools/orbital-velocity-period
/tools/hohmann-transfer
/tools/tsiolkovsky-delta-v
/tools/shaft-speed-critical
/tools/vibration-natural-freq
/tools/antenna-gain-beamwidth
/tools/gear-ratio

Sources

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