Rose Curve
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).
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.
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
where E(k) is the complete elliptic integral of the second kind, and the area of a petal is
| [画像: A_(petal)=(pia^2)/(4n). ] |
(4)
|
See also
Daisy, Dürer Folium, Epitrochoid, Limaçon Trisectrix, Maurer Rose, Quadrifolium, Starr Rose, TrifoliumExplore with Wolfram|Alpha
More things to try:
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." https://mathshistory.st-andrews.ac.uk/Curves/Rhodonea/.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.Referenced on Wolfram|Alpha
Rose CurveCite this as:
Weisstein, Eric W. "Rose Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RoseCurve.html