polar curves
Once you can name points by distance and direction, you can describe a whole curve by a rule that says how far out to go for each direction — an equation of the form r = f(theta). The shapes that pour out of such rules are some of the prettiest in all of geometry: circles, hearts, spirals, and many-petalled flowers, several of which have no tidy equation at all in ordinary x and y.
To picture a polar curve, let theta sweep around from 0 and, at each angle, mark the point at distance r = f(theta) from the pole. A few families recur. The simplest, r = a, is a circle of radius a centred at the pole (constant distance, every direction). A cardioid, r = a(1 + cos theta), traces a heart-shaped curve with a single dimple, swelling to 2a in one direction and pinching to 0 in the opposite one. A rose, r = a cos(n theta) or r = a sin(n theta), is a flower of petals: it has n petals when n is odd and 2n petals when n is even — a small surprise worth remembering. The spiral of Archimedes, r = a theta, winds outward at a steady rate because the distance grows in step with the angle.
These curves are not just decorative. Cardioids describe the pickup pattern of a directional microphone, roses and spirals model interference and growth, and conic sections themselves have clean polar equations centred at a focus, which is why polar form is the natural language of orbits. The thing to internalize is that the picture depends entirely on how r responds to theta — where r hits zero the curve passes through the pole, and where r is largest the curve reaches farthest out.
The rose r = 2 cos(3 theta) has n = 3, which is odd, so it draws exactly 3 petals (not 6), each reaching out to a maximum distance of 2 where cos(3 theta) = 1. A would-be sixth petal retraces a petal already drawn, which is why odd n gives n petals.
For the rose r = a cos(n theta), odd n gives n petals and even n gives 2n petals — a genuine surprise.
The petal count of a rose is the classic trap: r = a cos(n theta) has n petals for odd n but 2n for even n, because for odd n the curve is retraced on the second pass around.