Rhodonea / Rose Curves

r = cos(k·θ) traces a rose with k petals (k odd) or 2k petals (k even). Rational k = n/d creates beautiful multi-lobed patterns. Explore the full family from pure math to intricate flower geometries.

Numerator n 3 Denominator d 1 Amplitude offset A 0.0 Phase offset φ 0.0π

Color mode Show polar grid: Animate:

r = cos(3θ)

Presets

Trefoil (3) 4-petal Cinquefoil 7/2 rose 3/5 star 4/3 clover 6/5 rose Hybrid 16-petal 4-leaf Lemniscate-like 7/5 complex

Luigi Guido Grandi (1723) first described these curves as "Rhodonea" (rose). The number of petals follows: k odd → k petals, k even → 2k petals, k rational n/d (in lowest terms) → n petals if n·d is odd, 2n petals otherwise. The curve closes after d complete traversals of [0, 2πd]. Extension r = cos(kθ) + A adds a central ring connecting the petals.