A rose curve (or rhodonea, named by Guido Grandi around 1725) is a curve drawn in polar coordinates by the equation r = a·cos(kθ) (or the sine variant). The radius a sets the petal length and k controls how many petals appear.

Polar to Cartesian

Each point is plotted as x = r·cos(θ), y = r·sin(θ), where r = a·cos(kθ). When r goes negative the point flips to the opposite side of the origin — which is how even k produces extra petals.

Petal-count parity rule (integer k)

• If k is odd → the rose has k petals.
• If k is even → the rose has 2k petals.

For odd k the curve retraces itself over the second half-turn, so it closes after θ = π; for even k it needs the full θ = 2π.

Rational k = n / d

With k = n/d in lowest terms the curve no longer stops after one turn. It closes after θ = d·π when n·d is odd, and after θ = 2d·π otherwise, weaving denser, overlapping petals. As k approaches an irrational value the path never exactly closes and would fill the disc densely.

Maurer roses (optional)

Connecting sampled points of a rose with straight chords at a fixed angular step produces a Maurer rose — a striking lace-like pattern hidden inside the smooth rhodonea.