Owl: Cartesian coordinates ask how far east and how far north. Polar coordinates ask: how far from a lighthouse, and at what compass angle? The pair (r, θ) relocates the same plane with a different grammar.
Circles centered at the origin become r = constant. Rays from the origin become θ = constant. Roses and limaçon curves appear as simple formulas in r(θ) that look wild in x and y.
We'll convert points, plot basic loci, and face two misconceptions: r must be positive, and θ is always unique. Navigation by radius and angle is flexible — sometimes more than one pair names the same spot.