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

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

*r *= *a *sin(n*θ*), or

*r *= *a *cos(n*θ*)

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*= 2*n*-petalled, - Irrational
*n*= infinite petals.

- Odd
- 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 (x^{2} + y^{2} = 4*a*^{2}*x*^{2}*y*^{2} [2].

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

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 T_{n}(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].

## References

[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