Polar coordinates

The formula

Going from polar to Cartesian:

x = r·cosθ,    y = r·sinθ

Reading: walk distance r along a ray tilted θ from the +x axis, and these are the east and north components of where you end up.

Going back:

r = √(x² + y²),    θ = atan2(y, x)

Reading: r is straight Pythagoras. The angle comes from atan2, the two-argument arctangent — not atan(y/x), for reasons covered below.

The area element — where most polar mistakes live:

dA = r·dr·dθ

Reading: a patch of "polar graph paper" is not dr·dθ; it's a wedge whose width grows with radius. The extra r is the Jacobian of the transformation (DLMF 1.5.39 ^[9]), and forgetting it silently wrecks every integral you compute.

Arc length and enclosed area for a curve r(θ):

ds = √( r² + (dr/dθ)² )·dθ,    A = ½·∫ r² dθ

Reading: path length has a "sideways" part from the sweep and a "outward" part from the radius changing. The area formula sums thin pie slices. Check it on a circle r = R: A = ½·R²·2π = πR².

Velocity and acceleration of a moving point, in polar components:

v = ṙ·r̂ + r·θ̇·θ̂
a = (r̈ − r·θ̇²)·r̂ + (r·θ̈ + 2·ṙ·θ̇)·θ̂

Reading: four acceleration terms from two coordinates, because the unit vectors themselves rotate. −r·θ̇² is centripetal; 2·ṙ·θ̇ is Coriolis, the one nobody remembers until a slewing mechanism does something the Cartesian model didn't predict.

The complex-number version engineers use daily:

z = x + jy = |z|·e^(jθ)

Reading: polar form of a complex number. Multiplying two of them multiplies the magnitudes and adds the angles — the entire reason phasor arithmetic works.

Where you meet it

On the antenna range. A pattern measurement is gain versus angle at fixed range — data born polar. The pattern plot on the report is a polar chart with a dB radial axis, and reading beamwidth, sidelobe levels, and front-to-back ratio off it is a standard skill check for anyone new to the RF group.
At the radar console and in the tracking software. The radar measures range and azimuth directly; the PPI display is a polar plot with your antenna at the pole. The track filter, though, usually runs in Cartesian — so every measurement passes through the conversion above, and the measurement noise (tight in range, wide in angle) turns into a banana-shaped uncertainty region that a naive filter handles badly.
On the VNA and in every AC circuit calc. Impedance and reflection coefficient live in polar form: 50 Ω at −23° says magnitude and phase in one breath. The Smith chart is a polar plot of reflection coefficient dressed up with impedance gridlines.
In the rotating-machinery review. Turbine blades, robot joints, cam followers, centrifuge payloads — anything on a rotating frame gets its equations of motion written in r and θ, and the review-board question "did you include the Coriolis term?" comes straight from the acceleration formula above.

How it works

The coordinate pair is a range and a bearing. That's the whole idea, and it's why the system fits radiating and rotating problems: an antenna doesn't care about east and north, it cares about direction and distance. Symmetry decides the coordinate system. If the physics is the same in every direction from some center — a pattern, a pressure field around a shaft, a vibration mode of a disk — polar coordinates collapse a two-variable problem toward a one-variable problem. If the geometry is rectangular, stay Cartesian; forcing polar onto a box is self-inflicted pain.

The conversions are exact, but three traps hide in them:

The arctangent quadrant trap. atan(y/x) cannot tell (3, 4) from (−3, −4) — both give 53.13°, and the second point actually sits at −126.87°. The atan2(y, x) function exists to fix this; every serious language has it. Code that computes bearing with a bare atan works fine in test, then fails the first time a target crosses into the third quadrant. This is the single most common polar-coordinate bug in tracking and navigation code.
The missing Jacobian. Converting an integral to polar form means dx·dy → r·dr·dθ. Drop the r and you get a wrong answer with no warning. It also shows up physically: cells on a polar mesh grow with radius, so uniform steps in θ give you 175 m of cross-range spacing per 0.1° at 100 km range. Angle errors are cheap up close and expensive far out — the reason radar range accuracy and bearing accuracy are spec'd separately.
Degrees versus radians. Every calculus formula on this page assumes radians. Every compass, azimuth readout, and mechanical drawing assumes degrees. The conversion is 180° = π rad, and the failure mode is a plot or a servo command that's wrong by a factor of 57.3.

Two structural quirks are worth knowing rather than discovering. First, the representation isn't unique: (r, θ) and (r, θ + 360°) are the same point, and conventions allowing negative r add more aliases. Any code that differences two bearings must handle the wrap at ±180°, or a target crossing due south teleports across the display. Second, the pole itself is degenerate — at r = 0 the angle is undefined, and formulas with 1/r in them (the polar Laplacian, for one) need special handling there. Meshing software and finite-difference schemes on polar grids all carry patch logic for the center point.

The rotating-frame terms deserve one concrete number. A mechanism moving outward at 2 m/s along an arm rotating at 1 rad/s picks up a Coriolis acceleration of 2·ṙ·θ̇ = 4 m/s² sideways — about 0.4 g that a Cartesian free-body sketch never shows. That term is real load on real bearings, and it's the standard trap in dynamics of slewing cranes, extending booms, and pick-and-place arms.

Last habit worth building: on antenna pattern plots, remember the radial axis is usually dB with a floor. The center of the plot is not "zero signal" — it's whatever the plot floor is, often −40 dB. Sidelobes that look tiny on the paper can still be 1% of your radiated power, which is exactly the kind of thing an EMI review cares about.

History

The oldest polar curve on record belongs to Archimedes, who studied the spiral r = a·θ around 225 BC in a treatise called On Spirals, working out its tangents and areas without anything resembling modern coordinates ^[1]^[2]. The idea sat mostly idle for eighteen centuries. In 1635 Bonaventura Cavalieri — the indivisibles man, one of the fathers of integration — used polar-style reasoning to compute the area inside an Archimedean spiral, showing the first turn encloses exactly one-third of its circumscribing circle, and he's generally counted the first writer to employ the coordinates in earnest ^[3]^[4]^[5].

Newton took the next step. In Method of Fluxions, written around 1671 and published in 1736, he treated polar coordinates as a general way to fix any point in the plane — one option among several coordinate systems he cataloged for finding tangents ^[3]^[5]. Jacob Bernoulli went deeper: in a 1691 Acta Eruditorum paper he specified points by distance from a pole and angle from a polar axis, and derived the radius of curvature for curves given in polar form ^[3]^[5]. Bernoulli loved the logarithmic spiral enough to ask for one on his tombstone with the motto Eadem mutata resurgo — "though changed, I rise again the same." When he died in 1705, the Basel stonemason carved an Archimedean spiral instead. The wrong spiral is still there, on the cloister wall of Basel Cathedral ^[6]^[7].

The name arrived last. The term "polar coordinates" is credited to the Italian mathematician Gregorio Fontana and his eighteenth-century circle, and it reached English in George Peacock's 1816 translation of Lacroix's calculus text ^[3]^[8]. The extension to three dimensions was suggested by Clairaut and worked out fully by Euler ^[3]^[5] — which is where spherical coordinates, and every az/el gimbal spec in this town, come from.

Related tools

/tools/antenna-gain-beamwidth — beamwidth is an angular slice of a polar pattern
/tools/radar-horizon — the range limit on every PPI display
/tools/doppler-shift — driven by the radial component ṙ, the polar velocity term
/tools/series-rlc-impedance — impedance reported in polar form, magnitude and phase
/tools/vswr-return-loss — reflection coefficient magnitude, the radial coordinate on a Smith chart
/tools/convert-angle — the degrees/radians conversion that trap 3 is about

Sources

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