MISSION · Geometry
Polar Coordinates
A second address system, better suited to anything round
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The idea in one sentence
Give a point’s distance from the origin and the direction you’d face to see it, and you’ve located it just as precisely as any pair of \(x\) and \(y\).
\[(r, \theta)\]Why have two systems
Because the right coordinates make a problem easy and the wrong ones make it ugly.
A circle of radius 3 in Cartesian coordinates is \(x^2 + y^2 = 9\). In polar coordinates it’s \(r = 3\). That’s the entire equation: distance from the centre is 3, regardless of direction.
Anything built around a centre — radar, orbits, sound spreading from a speaker, a spiral shell — is simpler in polar form. Anything built on a grid — city blocks, spreadsheets, screens — is simpler in Cartesian.
Converting between them
The link is a right triangle, so it’s trigonometry:
\[x = r\cos\theta, \qquad y = r\sin\theta\] \[r = \sqrt{x^2 + y^2}, \qquad \theta = \tan^{-1}\!\left(\frac{y}{x}\right)\]One warning on that last one. The inverse tangent can’t distinguish opposite directions — \((1, 1)\) and \((-1, -1)\) give the same ratio — so it returns an angle in the right-hand half of the plane whatever you feed it. Check which quadrant the point is actually in and add \(180°\) if the calculator has put it in the wrong one.
Addresses aren’t unique
This is the real difference from Cartesian coordinates, and it catches people out.
A point has one \((x, y)\) and infinitely many \((r, \theta)\). Adding a full turn to the angle lands you back in the same place, so \((2, 30°)\), \((2, 390°)\), and \((2, -330°)\) are all the same point. A negative \(r\) is usually allowed too, meaning “go backwards through the origin”, giving yet more names.
And the origin is the extreme case: \(r = 0\) at any angle.
Curves that are hard to write otherwise
Polar equations produce shapes that are painful in Cartesian form:
- \(r = a\) — a circle.
- \(r = \theta\) — a spiral, winding outwards as the angle grows.
- \(r = a\cos(k\theta)\) — a rose, with \(k\) petals when \(k\) is odd and \(2k\) when it’s even.
- \(r = a(1 + \cos\theta)\) — a cardioid, heart-shaped, and the pickup pattern of most stage microphones.
Each of these is a couple of symbols in polar form and an unpleasant mess in \(x\) and \(y\).
Common mix-ups
- Trusting \(\tan^{-1}\) blindly. Check the quadrant.
- Mixing degrees and radians within one calculation.
- Assuming one point has one polar address. It has infinitely many.
- Reversing the pair. The distance comes first, then the angle.
Check yourself
Convert \((r, \theta) = (4, 60°)\) to Cartesian, and \((0, -5)\) to polar.
\(x = 4\cos 60° = 2\) and \(y = 4\sin 60° \approx 3.46\). For the second, \(r = 5\) and the point is straight down, so \(\theta = 270°\) — or \(-90°\), equally correct.
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