Rhodonea Curve (Rose Curve)

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A Rhodonea curve (also called a rosette, rosace, roseate curve or rose curve) is a plane curve shaped like a petalled flower.

Equations for the Rhodonea curve

five petal rose curve

Five petal rodonea curve.

The rhodonea curve is a sinusoid (a curve with the form of a sine wave) with the polar equation

r = a sin(nθ), or
r = a cos(nθ)

The two variants differ by a rotation of π/2.

  • The variable a controls the amplitude of the function; graphically, it controls how far from r = 0 the curve extends outwards. For example, r = 4*sin(4θ) means that the curve extends to y = 4.
  • The variable n controls the number of petals [1]:
    • Odd n = n-petalled.
    • Even n = 2n-petalled,
    • Irrational n = infinite petals.
  • The independent variable, θ, is the angle;
  • The dependent variable, r, is the radius as a function of θ.

The Cartesian equation is a lot less intuitive and a lot more complex. For example, the four petal quadrifolium (Rosace a quatre jeulles), has the polar equation r = a sin 2θ or the Cartesian equation (x2 + y2 = 4a2x2y2 [2].

The following video clip shows a = 1 with varying values for n.

Rhodonea curve
Watch this video on YouTube.

You can play around with the various values using this Desmos graph.

The curve can also be described by a pendulum in a plane orthogonal to its rotating axis, as n is large [3].

There is an interesting connection between the rhodonea curve and first kind Chebyshev polynomials; the polar equation for the rhodonea curve can be interpreted as Tn(cosθ), which is the definition of Chebyshev polynomials of the first kind [4].

History of the Rhodonea Curve

Luigi Guido Grandi (1671 to 1742), a mathematics professor at the University of Pisa, Italy, studied the curve in the 1700s and named it the rhodonea curve, which translates to “rose” [5].


[1] Rose. Retrieved February 25, 2022 from: https://archive.lib.msu.edu/crcmath/math/math/r/r382.htm
[2] Smith, D. (1958). History of Mathematics, Vol. II. Dover Publications.
[3] Pons, O. (2015). Analysis And Differential Equations. World Scientific Publishing.
[4] Integral Transforms and Operational Calculus.
[5] The Rose Curve. Retrieved February 25, 2021 from: https://mse.redwoods.edu/darnold/math50c/CalcProj/sp07/hanskatie/therose.ppt

Stephanie Glen. "Rhodonea Curve (Rose Curve)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/rhodonea-curve-rose/

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