Polar Coordinates

The polar grid

In Cartesian coordinates, you locate a point by going right x units and up y units. In polar, you point a ray from the origin at angle θ (measured from the positive x-axis, counterclockwise) and walk r units along it. The pair (r, θ) names the same point.

            θ = π/2
              │
   θ = π   ───┼───   θ = 0     <-- rays of constant θ
              │
            θ = 3π/2

   r=1: ─── circle of radius 1
   r=2: ─── circle of radius 2     <-- circles of constant r

The grid lines are concentric circles (constant r) and radial rays (constant θ). Cartesian grid lines are perpendicular straight lines; polar grid lines are perpendicular too, but they curve. This is what makes circular geometry simpler in polar — the grid already has circles in it.

Conversion between Cartesian and polar

The conversion formulas drop out of right-triangle trigonometry on the radial ray:

       y │       •  (x, y) = (r cos θ, r sin θ)
         ┤      ╱│
         ┤   r ╱ │ y = r sin θ
         ┤    ╱  │
         ┤   ╱θ  │
   ──────┼──────┼──── x
              x = r cos θ

Polar to Cartesian (always single-valued):

x = r cos θ
y = r sin θ

Cartesian to polar (with caveats):

r = √(x² + y²)
θ = atan2(y, x)         (full-quadrant arctangent)

Use atan2(y, x), not atan(y/x). The naive arctangent loses sign information — atan(−1/−1) and atan(1/1) both equal π/4, but the points (1, 1) and (−1, −1) are in opposite quadrants. atan2 takes both arguments and returns the correct angle in (−π, π].

Polar equations of common curves

Many curves with rotational or radial symmetry have remarkably compact polar equations.

Curve	Polar equation	Cartesian equation
Circle radius a, centered at origin	r = a	x² + y² = a²
Circle through origin, center (a, 0)	r = 2a cos θ	(x − a)² + y² = a²
Cardioid	r = a(1 + cos θ)	(x² + y² − ax)² = a²(x² + y²)
Lemniscate of Bernoulli	r² = a² cos(2θ)	(x² + y²)² = a²(x² − y²)
Archimedean spiral	r = aθ	(transcendental — no clean form)
Logarithmic spiral	r = a·e^(bθ)	(transcendental)
Rose with n petals (n odd)	r = a cos(nθ)	(degree-(n+1) polynomial)
Conic section, focus at origin	r = ℓ/(1 + e cos θ)	(quadratic in x, y)

Compare the row for the cardioid: r = a(1 + cos θ) versus (x² + y² − ax)² = a²(x² + y²). The polar form is a single transparent equation; the Cartesian form is a degree-4 implicit polynomial.

Why dA = r dr dθ — the Jacobian

To integrate over a region in polar, you need to know how a small (r, θ) "cell" maps to area in (x, y). Look at a small wedge between r and r + dr, between θ and θ + dθ:

                ╱
        ╱──────╱
      ╱      ╱ ← r dθ (arc length ≈ radius × angle)
    ╱──────╱ ← dr (radial step)
   ╱ ↗
  ╱ θ
 ┼─────────── x

   Cell area ≈ (radial step) × (arc length)
             = dr × r dθ
             = r dr dθ

The arc length r dθ is the key — it grows linearly with r. So polar cells get wider as you move outward. The factor r in dA = r dr dθ is the Jacobian determinant of the transformation:

J = | ∂x/∂r  ∂x/∂θ |  =  | cos θ   −r sin θ |  =  r cos²θ + r sin²θ  =  r
    | ∂y/∂r  ∂y/∂θ |     | sin θ    r cos θ |

Forgetting the r is the most common bug in polar integration. Computing ∫∫ f dr dθ when you mean ∫∫ f r dr dθ can be off by orders of magnitude over large regions.

Worked example — area of a circle

The area of a disk of radius a is most easily computed in polar:

A = ∫₀^(2π) ∫₀^a r dr dθ
  = ∫₀^(2π) [r²/2]₀^a dθ
  = ∫₀^(2π) a²/2 dθ
  = a²/2 · 2π
  = πa²

The familiar πr². Without the r in the area element, this integral would give 2πa — the circumference, not the area.

Worked example — Gaussian integral

The famous ∫_{−∞}^{∞} e^(−x²) dx = √π is impossible in elementary terms but trivial in polar:

I² = (∫e^(−x²) dx)(∫e^(−y²) dy) = ∫∫e^(−(x²+y²)) dx dy
   = ∫₀^(2π) ∫₀^∞ e^(−r²) · r dr dθ
   = ∫₀^(2π) [−e^(−r²)/2]₀^∞ dθ
   = ∫₀^(2π) (1/2) dθ = π

⇒ I = √π

The r in the area element converts e^(−r²) into e^(−r²) · r, whose antiderivative is elementary. This is why polar is the right tool for radially symmetric integrals.

Polar vs Cartesian — when each wins

	Cartesian (x, y)	Polar (r, θ)
Best symmetry match	Translational (rectangles, lines)	Rotational (circles, spirals)
Distance from origin	√(x² + y²)	r directly
Equation of a circle r = a	x² + y² = a² (implicit)	r = a (explicit)
Equation of a line through origin	y = mx (one slope)	θ = constant
Off-origin straight line	y = mx + b (clean)	r = b/(sin θ − m cos θ) (messy)
Area element	dA = dx dy	dA = r dr dθ
Coordinate singularity	None	Origin r = 0
Uniqueness of representation	One (x, y) per point	(r, θ + 2πk) all give same point

Where polar coordinates earn their keep

Kepler orbits

A planet around the Sun follows an ellipse with the Sun at one focus. In polar coordinates centered on the Sun:

r(θ) = ℓ / (1 + e cos θ)

One equation. The Cartesian form is a quadratic in x and y centered off the focus — comparatively unwieldy. Newton derived this from inverse-square gravity in Principia (1687); polar coordinates make the algebra tractable.

Antenna radiation patterns

Engineers describe how an antenna radiates with a polar plot of intensity vs angle. A dipole's pattern is r = sin θ; a directional array might be r = cos⁴(θ/2). The polar plot shows the beamwidth and side lobes at a glance.

Physics — central forces

Any force that points along the radial direction (gravity, Coulomb, harmonic spring centered at origin) is naturally written in polar. The conserved angular momentum L = mr²θ̇ is a polar quantity; angular-momentum conservation is invisible in Cartesian.

Image processing — log-polar transform

Mapping (x, y) ↦ (log r, θ) turns rotation about the origin into vertical translation, and zoom about the origin into horizontal translation. Pattern-matching algorithms exploit this for rotation- and scale-invariant detection.

Polar plots in statistics

Wind direction histograms (rose diagrams), circular statistics for time-of-day data, and biological orientation studies use polar plots because the angular variable is intrinsically circular — December and January are adjacent on a polar plot, but far apart on a linear axis.

Counterexamples and cautions

• The origin is singular. At r = 0, θ is undefined; (0, 0) corresponds to (0, θ) for every angle. A polar function must give the same value as r → 0 from every direction, or it's not continuous at the origin.
• (r, θ) is not unique. (r, θ) and (r, θ + 2π) name the same point, as do (r, θ) and (−r, θ + π) under the negative-radius convention. Naively summing or comparing polar coordinates without normalizing produces wrong answers.
• r = −1 is not "less than" r = 1. If you allow negative r, then (−1, 0) is the same point as (1, π) — distinct from (1, 0), but with the same |r|. Comparing radii without conventions agreed in advance is ambiguous.
• The Jacobian r is not optional. ∫∫ f(r, θ) dr dθ is meaningless as area; you need ∫∫ f(r, θ) r dr dθ for the integral to compute the area of the region under f.
• arctan is not enough. Computing θ = atan(y/x) drops sign info from x. (1, 1) and (−1, −1) both give atan(1/1) = π/4, but the second point is in quadrant III at 5π/4. Use atan2 in code.
• Curves like r = sin(2θ) trace different paths under sign conventions. If r ≥ 0 is required, you only see two petals of the rose; under the negative-radius convention, the curve reflects through the origin and traces all four petals. The picture changes with the convention.

Common mistakes

• Forgetting the r in dA. Most common polar bug. Area element is r dr dθ, not dr dθ.
• Using atan instead of atan2. Loses quadrant. Off by π in half the cases.
• Mixing r ≥ 0 with negative-radius conventions. Either is consistent, but mixing produces phantom 180° errors. Pick one and stick with it.
• Treating θ as a unique number. θ and θ + 2π are the same direction. When solving equations, expect multiple θ solutions per period.
• Misjudging the integration order. Inner integral over r typically depends on θ for non-circular regions: ∫_α^β ∫_0^{r(θ)} r dr dθ. Setting r-bounds independent of θ assumes the region is bounded by concentric circles.
• Using polar for problems with translational symmetry. A square or strip in polar is a mess of bounds. Polar shines for circles, spirals, sectors — not rectangles.
• Computing arc length without the (dr/dθ)² term. Polar arc length is ∫√(r² + (dr/dθ)²) dθ — both r and dr/dθ contribute. Using only ∫r dθ measures angular sweep, not curve length.