The Unit Circle

Chapter 6

Polar coordinates

Name a point by how far away it is and in which direction. The unit circle, stretched by r, turns that into x and y.

11 min read · 5 interactive figures

Drag the point. Its polar name is the distance r and the angle θ; its Cartesian name is x = r cos θ and y = r sin θ.

Polar coordinates name a point by its distance r from a fixed point, the pole, and the angle θ of the direction to it, measured counter-clockwise from the positive x-axis. To turn (r, θ) into ordinary coordinates, take the unit-circle point (cos θ, sin θ) in that direction and stretch it by r: x = r cos θ, y = r sin θ[1]. Everything else on this page, including the roses and spirals, follows from that.

One point, two names

Figure 1 Drag the point. Its Cartesian name is the blue and vermilion bars (x and y); its polar name is the length of the radius (r) and the angle (θ). The small gray dot is (cos θ, sin θ) on the dashed unit circle: the point is that dot, pushed out to distance r.#

Going from polar to Cartesian is the unit circle and a multiplication:

x = r cos θ     y = r sin θ(6.1)

Going back, the distance is Pythagoras, r = √(x² + y²). The angle is where people slip. It is tempting to write θ = tan⁻¹(y/x), and that is right only in quadrants I and IV. The inverse tangent always answers with an angle between −90° and 90°, because it cannot tell (−1.5, 2) from (1.5, −2): both have y/x = −4/3. For the point (−1.5, 2) a calculator gives tan⁻¹(−4/3) ≈ −53.1°, which points into quadrant IV, the opposite direction. The point is in quadrant II, so add 180° to get 126.9°. Spreadsheets and programming languages provide a two-argument function, usually called atan2(y, x), that does this bookkeeping for you; the figures on this site use it.

Unlike Cartesian coordinates, polar names are never unique. Adding a full turn to θ changes nothing, so (r, θ) and (r, θ + 2π) are the same point. And a negative distance is allowed by convention: (−r, θ) means go distance r in the direction opposite to θ, which is the same as (r, θ + π). The pole itself has r = 0 and any angle at all. The sentence under the figure lists some of the other names for the point you choose.

Polar graphs: plotting r = f(θ)

A polar equation gives the distance as a function of the angle. To draw it, imagine a ray turning about the pole like the hand of a clock running backwards. At each angle, put a dot at the distance the equation gives. The trail of dots is the curve. Beside it, draw the same function the ordinary way, with θ across and r up. The two drawings carry the same information, and comparing them is the quickest way to understand a polar curve.

Figure 2 Press Trace, or drag the θ slider. On the left the ray turns and the point sits at distance r(θ) along it; on the right is the graph of r against θ. Start with the cardioid, r = 1 + cos θ, then try the other curves in the list.#

The cardioid shows the method clearly. At θ = 0, r = 2, the farthest point. As the ray turns, r shrinks, and at θ = π it is zero: the curve comes into the pole with a cusp, the point of the heart. The right-hand graph is a cosine wave lifted by 1, never negative, which is why the cardioid never crosses the pole to the other side.

Roses, and what negative r does

Question 1 A prediction about the rose r = cos 2θ.#
Figure 3 The rose r = cos(kθ). Press Trace and watch each hump of the right-hand graph become a petal. While r is negative the bar turns blue and the point is drawn on the opposite ray. Change k and count petals for odd and even values.#

The petal count follows a rule that looks arbitrary until you watch it: the rose r = a cos(kθ), with k a whole number, has k petals if k is odd and 2k petals if k is even[1]. Negative r explains it. Each hump of cos kθ, up or down, sweeps out one petal; the downward humps are drawn on the opposite side of the pole. When k is even, the opposite side is empty and the negative humps add new petals. When k is odd, the opposite side is exactly where an existing petal already sits, so the second half-turn retraces the first: the curve is complete at θ = π, with only k petals.

A small catalog of polar graphs

Each of the classic polar curves is in the list in the figures. A few are worth knowing by their equations[1]:

  • Circle through the pole: r = 2a cos θ. Multiply both sides by r and use r² = x² + y² and r cos θ = x: you get x² + y² = 2ax, a circle of radius a centered at (a, 0). It is complete after half a turn.
  • Cardioid and limaçons: r = a(k + cos θ). At k = 1 it is the cardioid. For k < 1, r goes negative for a while and the curve grows a small inner loop; for k between 1 and 2 it has a dimple; at 2 or more it is convex.
  • Lemniscate: r² = a² cos 2θ, a figure eight. It exists only where cos 2θ ≥ 0, so the figure leaves gaps in the right-hand graph where there is no point at all.
  • Archimedean spiral: r = aθ. The distance grows steadily with the angle, so successive turns are evenly spaced, like the groove of a record.
  • Conic sections: r = ℓ/(1 + e cos θ), with the pole at a focus. The number e, the eccentricity, decides the shape: a circle at e = 0, an ellipse for e between 0 and 1, a parabola at exactly 1 and a hyperbola beyond[2]. This one gets a section of its own below.

Sketching a polar graph by hand

The figures do the plotting, but on a test you will do it with a pencil, and the method is the one the figures use, slowed down. Three habits make it quick.

Make a short table at the special angles. For r = 1 + 2 cos θ (the limaçon with k = 1/2 from the list, doubled in size), the values at 0, π/3, π/2, 2π/3 and π are 3, 2, 1, 0 and −1. The unit circle supplies every cosine in that row, which is why the special angles are worth knowing. Plot each point by turning to the angle and walking out the distance, and remember that a negative distance is walked backwards, through the pole.

Look for the zeros of r. Where r = 0 the curve passes through the pole, and it passes through heading in the direction of that angle. For 1 + 2 cos θ that happens at 2π/3 and 4π/3, and between them r is negative: that stretch is the small inner loop, drawn on the side of the pole facing angle 0.

Use symmetry to halve the work. If replacing θ with −θ leaves the equation unchanged, the graph is symmetric about the horizontal axis, and any equation built only from cos θ passes this test because cos(−θ) = cos θ. If replacing θ with π − θ leaves it unchanged, the graph is symmetric about the vertical axis, which is the test that sin θ passes. These tests are sufficient, not necessary: a curve can be symmetric even when the test fails, because the same point has more than one polar name[1]. established The figure with the right-hand r–θ graph is the same table, filled in at 900 angles instead of five.

The area inside a polar curve

Polar coordinates make some areas easy that are awkward in x and y, and the reason is again the circle. A thin slice of the region between angle θ and θ + Δθ is almost a sector of a circle of radius r, and a sector of angle Δθ has area ½r²Δθ. That formula needs Δθ in radians, for the reason given in Chapter 2: the arc of the sector is rΔθ. Adding up the slices gives the area of the region swept as θ runs from α to β[2]:

A = 12 ∫αβ r² dθ(6.2)

For the cardioid r = 1 + cos θ traced once, r² = 1 + 2 cos θ + cos²θ. Over a full turn the middle term averages to zero and cos²θ averages to one half, so the integral is 2π + π = 3π and the area is 3π/2, exactly one and a half times the area of the unit circle. established The squared r also explains a trap with roses: because r² ignores the sign, integrating over a range where the curve retraces itself counts each petal twice.

Polar graphs in the world

Polar graphs are the natural picture whenever something depends on direction. Two examples show how far the same few equations reach.

Microphone patterns

Every microphone spec sheet has a polar graph showing how sensitive the microphone is to sound arriving from each direction. The common patterns are exactly the limaçons in the figures above, with names of their own. A manufacturer’s list gives them as ρ = 1 + 0 cos θ for an omnidirectional microphone, ρ = 0.5 + 0.5 cos θ for a cardioid, ρ = 0.37 + 0.63 cos θ for a supercardioid, ρ = 0.25 + 0.75 cos θ for a hypercardioid and ρ = cos θ for a figure-of-eight[3]. documented The cardioid, the heart-shaped curve this chapter started with, is the classic stage microphone: full sensitivity in front, half at the sides, and in theory none from behind, where the singer’s monitor speaker sits.

The negative values of r matter here. For a supercardioid, 0.37 + 0.63 cos θ is negative for sounds from behind, which means the microphone picks them up with inverted polarity. A spec sheet plots the size of the response, |ρ|, so it shows a small rear lobe behind the microphone. Plotted with signed r, the way a mathematician (and the figure above) would, the same lobe lands in front, as the inner loop of a limaçon. Choose the limaçon in Figure 2 and set k to 0.6 to see it. Same numbers, two conventions, and a good reason to check which one a graph uses.

Orbits

Kepler’s first law says that a planet moves on an ellipse with the Sun at one focus. Put the pole at that focus and the ellipse has the polar equation r = ℓ/(1 + e cos θ), which is why astronomers write orbits in polar form[2]. The planet is closest to the Sun at θ = 0, where r = ℓ/(1 + e), and farthest at θ = π, where r = ℓ/(1 − e).

Figure 4 A conic with a focus at the pole. Drag the eccentricity e (the parameter slider): 0 gives a circle, values up to 1 a longer and longer ellipse, 1 a parabola that never closes, and more than 1 one branch of a hyperbola. Press Trace to follow a planet around its orbit.#

Earth’s orbit has eccentricity 0.0167, so close to a circle that you could not tell the difference by eye. Even so the distance changes by about 5 million kilometers through the year: 147.1 million km at perihelion, the nearest point, and 152.1 million km at aphelion, the farthest[4]. established The ratio of the two is (1 − e)/(1 + e) ≈ 0.967, which you can check from the polar equation without knowing ℓ.

Where polar coordinates came from

The idea of locating points by distance and direction is old, but its use as a coordinate system is a seventeenth-century development. In a lecture at the 1950 International Congress of Mathematicians, the historian of mathematics Julian Coolidge traced it back to Bonaventura Cavalieri, who used polar-style coordinates to find the area inside an Archimedean spiral by relating it to the area outside a parabola. Coolidge credited Isaac Newton as the first to treat them as a way of fixing any point of the plane, and Jacob Bernoulli with going as far as a formula for the curvature of a curve given in polar form[5]. documented

The deepest use of the idea came from Euler. Write a point of the plane as a complex number, x + iy. In polar form that is r(cos θ + i sin θ), and in 1748, the same year as the radius-1 circle of the Introductio, Euler gave the formula eiθ = cos θ + i sin θ[6]. documented The point going around the unit circle at the start of this site is the number eiθ, and multiplying by it is a rotation. That is the start of another story; this one began with a point on a circle, and it ends there.