🌹 Rose Curves
Rhodonea — r = a·cos(kθ)
r = a·cos((3/1)·θ)
Petals: 3
Function
Presets
Shape k = n / d
Animation
Color
Controls
Stats
Petals
3
Closes at
1π
Progress
0%
k = n/d
3
Info & Theory

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.

Drag — rotate · Scroll — zoom

About Rose Curves

Written by MySimulator Team · Reviewed by MySimulator Editorial Review

Last updated: 5 July 2026

Rose curves (rhodonea curves) are polar curves defined by r = a·cos(kθ) or r = a·sin(kθ), named by Italian mathematician Guido Grandi in 1728. The parameter k determines the petal structure: when k is a positive odd integer the curve has exactly k petals; when k is a positive even integer it has 2k petals. For irrational or rational non-integer k = n/d (with n, d coprime), the curve never closes in one revolution and must be traced through d full turns of θ to complete, producing exotic multi-layered roses with n petals if n+d is odd, or 2n petals if both n and d are odd.

Watch each petal traced out smoothly as θ advances, observe the symmetry change as k crosses an integer, and explore extended roses with rational k = n/d to see how the number of loops and their overlap pattern depend on the arithmetic of the fraction.

Frequently Asked Questions

Why does an even k give 2k petals but odd k gives only k petals?

For odd k, the n negative half-petals (where cos(kθ) < 0) retrace the same paths as the positive half — so the curve completes with k distinct petals after one revolution (θ: 0 → 2π). For even k, the negative lobes fall in different angular positions from the positive lobes and do not overlap, yielding 2k distinct petals. This parity dependence is a consequence of the function cos(kθ) having period 2π/k and the symmetry properties of the polar coordinate conversion.

What happens with rational k = n/d?

When k = n/d (in lowest terms), the curve r = a·cos(n/d · θ) is periodic with period 2πd: the curve must be traced for d full revolutions to close. The number of petals is n if (n+d) is odd, or 2n if both n and d are odd. For example, k = 3/2 gives a 3-petal rose traced in 2 revolutions; k = 2/3 gives 4 petals in 3 revolutions. Irrational k produces a quasiperiodic curve that densely fills an annular region without ever closing.

What is the area enclosed by a rose curve?

For r = a·cos(kθ) with integer k, the total area is πa²/2 regardless of the number of petals — each petal has area πa²/(4k) for odd k (k petals) and πa²/(8k) for even k (2k petals), and the totals always add to πa²/2. This counterintuitive result means a 100-petal rose encloses the same total area as a 3-petal rose of the same amplitude a — the petals simply get thinner as k increases.

Who first studied rose curves and when?

Rose curves were introduced by Guido Grandi (1671–1742), an Italian monk and mathematician, in his 1728 paper "Flores geometrici." He named them "rhodonea" from the Greek for rose. The curves attracted attention from Luigi Guido Grandi's contemporaries including Leibniz, and they became a standard example in 18th-century texts on analytic geometry and polar coordinates.

How are rose curves related to Lissajous figures?

A Lissajous figure traces x = A sin(nωt + φ), y = B sin(mωt) — the trajectory of a point with two perpendicular oscillations of frequency ratio n:m. Converting to polar coordinates, a Lissajous figure with equal amplitudes and a phase shift of π/2 coincides exactly with a rose curve r = cos(n/m · θ). This means rose curves can be produced mechanically on an oscilloscope or harmonograph by driving two perpendicular oscillators at a rational frequency ratio.

Can a rose curve have a fractional number of petals?

With rational k = n/d the curve has an integer number of petals, but they may overlap, creating a layered appearance rather than distinct separate petals. With k slightly irrational, the curve never closes; instead it densely fills the disc of radius a, weaving a tight quasi-periodic mesh. The transition from closed rose to dense filling as k changes from rational to irrational is an example of how number theory (rational vs. irrational) has dramatic geometric consequences.

What is the curvature at the tips of rose petals?

At the tip of a petal (where r = a, θ = 0 for r = a·cos(kθ)), the curvature κ = 1/(ak) — larger k means more tightly curved petal tips. Near the origin, each petal pinches to a cusp (for integer k the origin is a node passed through 2k times), and the curve's local behaviour near the origin follows r ≈ a·k·|θ − θ₀|, producing sharp cusps. This cusp structure is exploited in antenna array design where rose-like beam patterns are desired.

Do rose curves appear in physics or engineering?

Yes — in several guises. Antenna radiation patterns sometimes approximate rose curves: a dipole antenna produces a two-petal toroidal pattern, and phased arrays can synthesise multi-petal directional beams. In quantum mechanics, the angular part of hydrogen atom orbitals (spherical harmonics |Ylm|²) plotted in polar form resemble rose curves, with l and m controlling petal number and shape. Rose-like patterns also appear in the polar plots of directional microphone sensitivity and in optical diffraction through multi-slit gratings.

What is the arc length of a rose curve?

The arc length of one petal of r = a·cos(kθ) is L = ∫₀^(π/2k) √(r² + (dr/dθ)²) dθ = a∫₀^(π/2k) √(cos²(kθ) + k²sin²(kθ)) dθ. This integral has no elementary closed form in general but can be expressed in terms of elliptic integrals. For k = 1 (one full petal = circle), L = πa exactly. For large k, each thin petal has arc length approximately 2a, so the total length grows as 2ak (odd k) or 4ak (even k) — linearly with k.

Can rose curves be generalised to 3D?

Yes. Rotating a 2D rose curve about an axis of symmetry produces a 3D rose surface; alternatively, spherical harmonics generalise rose curves to the surface of a sphere. More exotic generalisations replace cos(kθ) with cos(kθ)·sin(φ) in spherical coordinates, producing rose-like polyhedra on the sphere. These appear in the study of resonant orbital patterns (orbital resonance in celestial mechanics produces rose-like Lissajous orbit figures in rotating reference frames).

How does shifting from cosine to sine change the rose?

Replacing cos with sin rotates the entire curve by π/(2k): r = a·sin(kθ) is simply r = a·cos(k(θ − π/(2k))) — the same petal shape rotated so that the first petal points at angle π/(2k) from the x-axis rather than along it. The petal count rules remain the same. Combining both, r = a·cos(kθ) + b·sin(kθ) = √(a²+b²)·cos(k(θ − arctan(b/a))), which is still a rose with the same petal count but amplitude √(a²+b²) and a phase-rotated orientation.