Polar Coordinates and Polar Curves
Locate a point by distance and angle instead of two displacements, convert between the two systems, and sketch curves that are simpler in polar form.
What a learner can do afterwards
- Convert a point and an equation between Cartesian and polar form
- Sketch a circle, a spiral and a rose curve from a polar equation by tracking the radius as the angle turns
- Say which shapes gain from polar form and which lose, with a reason
1 · Read
Polar coordinates name a point by how far out and which way round: radius r and angle theta. You convert with x equals r cos theta and y equals r sin theta, and going back you use r squared equals x squared plus y squared. One point owns many names, since adding a full turn or flipping the radius lands back in place.
Plot by walking the angle first, then stepping out along the radius. The point (2, 90 degrees) gives x equals 0 and y equals 2, so it is (0, 2) in Cartesian form. The point (6, 180 degrees) gives x equals 6 times minus 1, which is minus 6. A negative radius steps backwards through the origin, which surprises beginners exactly once.
To sketch a polar curve, track r as theta turns and watch where r hits zero, peaks, or dives negative. A fixed radius like r equals 2 traces a circle centred at the origin, r equals theta winds a spiral outward, and r equals 4 cos(3 theta) draws a 3 petal rose, since odd n gives n petals. Centred circles collapse beautifully, while a shifted line like x equals 5 turns into the clumsy r equals 5 sec theta.
Judge the winner by simplicity. Reach for polar form when rotation or roundness dominates, and keep x and y when the curve is shifted or straight.
Walk the angle, step out the radius, and pick the system that makes your curve look simplest.
2 · Watch
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Where it sits
Learn first
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Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.