How to graph polar equations

Polar coordinates describe a point by how far out it is and which way it faces. That single change turns equations that would be unwieldy in Cartesian form into something you can write in a handful of characters. r = 1 + cos θ is a perfect heart.

This guide covers the coordinate system, the famous curves and how to recognise them, the conversions in both directions, and every control in the free polar graphing calculator. No download and no account.

Every point is written as (r, θ): how far out, and at what angle.

The pole is the centre, the polar equivalent of the origin. The polar axis is a ray pointing right from the pole, and all angles are measured from it exactly as the positive x-axis would be. r is the straight-line distance from the pole to the point. θ is the turn from the polar axis, counter-clockwise for positive values, clockwise for negative, given in radians or degrees.

Two things about polar coordinates surprise people, and both are worth getting straight before you plot anything.

A negative r is allowed, and it means the opposite direction. The point (−2, 30°) is not somewhere near 30 degrees. You face along the 30 degree ray, then walk two units backwards, landing at (2, 210°). The two notations describe the identical physical spot. This is exactly why some polar curves appear to reach across the pole and draw loops on the far side, because the equation is producing negative radii for part of its range.

Take the point P(−6, −3π/4). The angle is 135 degrees measured clockwise. The radius is negative, so you flip 180 degrees, ending up on the 135 degree counter-clockwise ray at distance 6, which in Cartesian terms is (−3√2, 3√2).

Every point has infinitely many names. (5, 0) and (5, 2π) are the same location, because a full turn brings you back. Add another 2π and you are there again. In Cartesian coordinates each point has exactly one address, but in polar coordinates it has endless ones, and that has real consequences when you look for intersections.

Type r = f(θ) into a sidebar row. The leading r = is optional, so 1 + cos(t) works just as well as the full form. Use θ or t for the variable, since both are accepted and t is easier to type.

The constants pi, e and tau are built in, along with the usual function set. Anything the calculator cannot parse gets a wavy underline rather than failing silently. If Greek letters are awkward on your device, open the math keyboard, which carries thirty keys with θ in the position x occupies on the graphing calculator.

The background is a genuine polar grid, with concentric rings for radius and spokes every 30 degrees for angle. Radius values are read from the number line along the axes, and the angle labels change with your mode: fractions of pi in radians, degrees in degrees. Grid and Labels toggle independently, so you can strip the display back for an export.

A segmented toggle switches between DEG and RAD, and it converts your range as it goes. Flip to degrees while showing 0 to 2π and the range becomes 0 to 360. Match it to whatever your problem uses.

Two inputs set the start and end of θ, deciding how much of the curve gets drawn. In polar this matters far more than the equivalent setting does elsewhere. Cut a spiral off at 2π and you see one loop of something infinite. Draw a three-petal rose all the way to 2π and it traces itself twice for no benefit.

Trace is the most useful control on the page. Hover anywhere and you get θ, r and (x, y) simultaneously, with the nearest point on the curve marked. That is a live polar to rectangular conversion, which makes the conversion section below far less abstract.

Key points marks two things automatically: a dot at each curve's maximum radius, and a gold marker at every pole crossing where r = 0. Those crossings are exactly what you would otherwise solve for by hand, and they are the boundaries between petals and loops. The table of values view is the digital version of the by-hand table described next.

You can add, edit inline, delete, recolour and hide several curves at once. Comparing a cardioid against a limaçon in one view is the fastest way to understand the difference between them.

Drag to pan, scroll to zoom toward the cursor, or use the plus and minus buttons and Reset view. Undo and redo cover sixty steps with Ctrl+Z and Ctrl+Y.

Copy link encodes the whole state into a URL, Save file writes JSON, Open file restores it, and PNG export produces an image. The Examples dropdown loads a circle, cardioid, three and four petal roses, a limaçon, a lemniscate, a spiral and a combined view.

All four shapes come from the same equation, and the ratio a/b decides which one you get. Below 1 you get an inner loop. At a = b you get a cardioid, the boundary case where the loop shrinks to a single cusp. Between 1 and 2 it is dimpled, the heart having lost its point. At 2 or above it is convex, a smooth off-centre oval.

That is worth internalising, because it means a cardioid is not a separate species. It is the exact moment a limaçon's inner loop disappears.

r = sin(3θ) gives three petals and r = sin(2θ) gives four. The rule looks arbitrary until you watch the tracing.

With an odd n, the curve completes the entire figure as θ runs from 0 to π. Carry on to 2π and it simply retraces the petals it already drew, because the negative radii in the second half land on top of the positive ones from the first. You see n petals. With an even n, nothing overlaps, so each half of the rotation produces its own fresh set and you finish with 2n.

Worth doing by hand once. At θ = 0, r = 0, so it starts at the pole. At θ = π/2, r = 3, the maximum. At θ = π it is back to 0. Between π and 2π, sin θ is negative, so every radius is negative and plots opposite, landing exactly on the circle already drawn. The result is a circle of diameter 3 sitting above the polar axis, complete after just half a turn.

For r = a(1 + cos θ), symmetry about the horizontal axis means you can integrate from 0 to π and double it. Expanding the bracket and using the half-angle identity gives the result above.

Radar and air traffic control. A radar screen is a polar grid, and not by coincidence. Aircraft are reported as a distance and a bearing, such as twenty kilometres out on bearing 045, because that is immediately actionable in a way a coordinate pair is not. The sweeping line is θ rotating through 360 degrees, and each blip's distance from the centre is its r.

Microphone and antenna patterns. The cardioid microphone is named directly after the curve. Its sensitivity follows r = 1 + cos θ, fully open at the front where θ is 0, and completely dead behind at 180 degrees where r drops to zero. That null is why a cardioid mic rejects the sound of the room behind it, and antenna engineers read radiation strength off exactly the same kind of plot.

Spirals in nature. Nautilus shells, hurricane rainbands and galaxy arms all approximate r = a·e^(bθ). Sunflower seed heads use a variation, with seeds sitting at a radius proportional to the square root of their index and the angle advancing by roughly 137.5 degrees each time, the golden angle. Two symbols produce one of the densest packings known.

Navigation and bearings. Sailors and pilots work in bearing plus distance, which is polar by another name. Converting that into a chart's grid reference is precisely the conversion above, done constantly and usually by a machine.

Planetary orbits. Kepler found that orbits are ellipses with the sun at one focus, which in polar form is r = p/(1 + e cos θ). Astronomers keep it in polar because θ, the true anomaly, is the natural variable for something going round. The eccentricity tells you the shape: 0 is a circle, below 1 an ellipse, exactly 1 a parabola, above 1 a hyperbola. Earth's is 0.0167, very nearly circular.

Honeybees. A returning forager dances the direction and distance of food relative to the hive, which is polar coordinates communicated by an insect. Karl von Frisch decoded it and won a Nobel Prize for it in 1973.