Calc III · §10.3 polar curves — rose curves

Why does r = cos(2θ) have four petals?

Watch the pen sweep: when r > 0 the point sits at angle θ, but when r < 0 it flips to the opposite side (θ+π) — and that flip is what draws the extra petals. Each hump of the cosine on the right is one petal on the left.

Polar plane — the pen draws the rose (x = r cosθ, y = r sinθ)

r versus θ — each hump is one petal; red = r is negative

Positive r keeps the direction; negative r flips it. That's the whole trick. 🌹