TOPICS
Search

Rose Curve


Rose

A rose curve, also called Grandi's rose or the multifolium, is a curve which has the shape of a petalled flower. This curve was named rhodonea by the Italian mathematician Guido Grandi between 1723 and 1728 because it resembles a rose. The polar equation of the rose is generally given as

 r=acos(ntheta)
(1)

(e.g., Lawrence 1972, p. 175; Ferréol; illustrated above) or by the version rotated by 90 degrees,

 r=asin(ntheta)
(2)

(MacTutor). The sine version has the advantage that roses with odd n have a petal oriented vertically (up or down depending on n), whereas the cosine orientation gives a petal oriented to the right.

If n is odd, the rose is n-petalled. If n is even, the rose is 2n-petalled.

The curve is algrebraic iff n=p/q is rational, with degree p+q when pq is odd and 2(p+q) when pq is even. The following table gives the algebraic forms for integer n-peteled roses r=asin(ntheta).

nequation
1x^2-ay+y^2
2x^6-4a^2x^2y^2+3x^4y^2+3x^2y^4+y^6
3x^4-3ax^2y+2x^2y^2+ay^3+y^4
4x^(10)-16a^2x^6y^2+5x^8y^2+32a^2x^4y^4+10x^6y^4-16a^2x^2y^6+10x^4y^6+5x^2y^8+y^10
5x^6-5ax^4y+3x^4y^2+10ax^2y^3+3x^2y^4-ay^5+y^6
RoseRational

If n=p/q is a rational number, then the curve closes at a polar angle of theta=piqm, where m=1 if pq is odd and m=2 if pq is even.

RoseIrrational

If n is irrational, then there are an infinite number of petals.

The rose curve is a special case of the hypotrochoid with h=a-b, giving a rose with scale a^'=2(a-b) and petal parameter n=a/(2b-a).

The following table summarizes special names gives to rose curves for various values of n.

The arc length of a single petal is

 s_(petal)=(2aE(sqrt(1-n^2)))/n,
(3)

where E(k) is the complete elliptic integral of the second kind, and the area of a petal is

 A_(petal)=(pia^2)/(4n).
(4)

See also

Daisy, Dürer Folium, Epitrochoid, Limaçon Trisectrix, Maurer Rose, Quadrifolium, Starr Rose, Trifolium

Explore with Wolfram|Alpha

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 223-224, 1987.Ferréol, R. "Rose." https://mathcurve.com/courbes2d.gb/rosace/rosace.shtml.Hall, L. "Trochoids, Roses, and Thorns--Beyond the Spirograph." College Math. J. 23, 20-35, 1992.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 175-177, 1972.MacTutor History of Mathematics Archive. "Rhodonea Curves." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Rhodonea.html.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 330, 1958.Wagon, S. "Roses." §4.1 in Mathematica in Action. New York: W. H. Freeman, pp. 96-102, 1991.

Cite this as:

Weisstein, Eric W. "Rose Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RoseCurve.html

Subject classifications