Rose Curve

rose curve
Rose curves as defined by “r==Cos[p/q*θ]”. The fraction on the upper right corner indicates the parameter p/q.

Mathematica Notebook for This Page.


Studied by Guido Grandi around 1723.

aka rhodonea


Rose describes a family of curves. Roses are probably historically defined as the pedals of epi/hypocycloids with respect to their centers, which has the polar form r==Cos[p/q*θ], where p and q are relatively prime. Not all possible rational p/q in r==Cos[p/q*θ] occur as pedals of epi/hypocycloids, but we can define rose to be the curve r==Cos[p/q*θ], where p/q is any rational. The curve has loops that are symmetrically distributed around the pole. The loops are called petals or leafs.

If p and q are both odd, it has a period of π*q with p petals, otherwise the period is 2*π*q and has 2*p petals. In particular, r==Cos[p/1*θ] has p petals and periodicy π if p is odd, else it has 2*p petals with period 2*π. The period of the curve can be derived by studying the period of the parametric form Cos[p/q*t]*{Cos[t],Sin[t]}. In particular, analying the period of Sin[p/q*t]*Sin[t]. Similarly, the number of petals can be inferred. The curve r==Cos[r*θ] where r is irrational, is non-periodic.

Roses are shaped like a airplane propeller when q==1. If q is not 1, the “propeller blades” are fat and overlap, which alludes to its name “rose”.

The rose r==Cos[3*θ] is called Trifolium (having 3 petals), and r==Cos[2*θ] is called Quadrifolium (having 4 petals). The pedals of a deltoid with respect to one of its cusp, vertex, or center is called a folium, bifolium, and trifolium respectively. They are called n-foliums because the curves have n loops. However, folium and bifolium are not roses. Only the trifolium is a rose. (*XahNote: verify if the curve historically called “folium” is the 1 petalled rose r==Cos[1/3*θ].*)


Polar equation: r==Cos[p/q*θ].

Cartesian equation for a 4-pedaled rose r==Cos[2*θ] rotated by 2*π/8 is (x^2+y^2)^3==4*x^2*y^2.

list all equation in rect coord for n-leafed rose. Proof if for some n its not algebraic.


Rose as Hypotrochoid

Roses are special cases of hypotrochoids. (See: Curve Family Index)

Roses are probably also epitrochoids. Prove or disprove it. Also, epitrochoid and or hypotrochoids has double generation; check how it applies to rose.

Pedal, Radial, Epycycloid, and Hypocycloid

The pedal and radial of a epi/hypocycloid with respect to their centers are roses. The following image epiHypocycloidEpiPedal.png shows 40 epicycloids (blue) and their pedals (red). See epi/hypocycloid page for more detail and illustration.

“Real” roses

real rose
Real Rose Curve
rose curve
This animation shows r == Sin[c*θ] with c going from 2 to 3. It shows what happens when c is irrational and how a 4 leafed rose transforms to a 3 leafed rose smoothly. When c is irrational, the curve fills a circle as θ goes to infinity.

Cotes's Spiral

Rose inverts to a curve called Cotes's Spiral.

rose curve
Rose inverts to a curve called Cotes's Spiral.

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

The MacTutor History of Mathematics archive

Rose (mathematics).

If you have a question, put $5 at patreon and message me.

Plane Curves


  1. Conic Sections
  2. Parabola
  3. Hyperbola
  4. Ellipse
  5. Cissoid
  6. Conchoid
  7. Quadratrix
  8. Archimedean Spiral
  9. Equiangular Spiral
  10. Lituus
  11. Cornu Spiral


  1. Epitrochoid
  2. Hypotrochoid
  3. Epicycloid and Hypocycloid
  4. Rose Curve
  5. Astroid
  6. Deltoid
  7. Nephroid
  8. Cardioid
  9. Trochoid
  10. Cycloid

Calculus Era

  1. Cassinian Oval
  2. Cross Curve
  3. Folium of Descartes
  4. Piriform
  5. Semicubic Parabola
  6. Tractrix
  7. Trisectrix
  8. Trisectrix of Maclaurin
  9. Lemniscate of Bernoulli
  10. Lemniscate of Gerono
  11. Limacon Of Pascal
  12. Witch of Agnesi
  13. Sine Curve
  14. Catenary
  15. Bezier Curve


  1. Caustics
  2. Cissoid
  3. Conchoid
  4. Envelope
  5. Evolute
  6. Involute
  7. Geometric Inversion
  8. Orthoptic
  9. Parallel Curve
  10. Pedal Curve
  11. Radial Curve
  12. Roulette

Math of Curves

  1. Geometry: Coordinate Systems for Plane Curves
  2. Coordinate Transformation
  3. Vectors
  4. Naming and Classification of Curves
  1. Cusp
  2. Curvature