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
(1)
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(e.g., Lawrence 1972, p. 175; Ferréol; illustrated above) or by the version rotated by ,
(2)
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(MacTutor). The sine version has the advantage that roses with odd have a petal oriented vertically (up or down depending on ), whereas the cosine orientation gives a petal oriented to the right.
If is odd, the rose is -petalled. If is even, the rose is -petalled.
The curve is algrebraic iff is rational, with degree when is odd and when is even. The following table gives the algebraic forms for integer -peteled roses .
equation | |
1 | |
2 | |
3 | |
4 | |
5 |
If is a rational number, then the curve closes at a polar angle of , where if is odd and if is even.
If is irrational, then there are an infinite number of petals.
The rose curve is a special case of the hypotrochoid with , giving a rose with scale and petal parameter .
The following table summarizes special names gives to rose curves for various values of .
curve | |
1/3 | limaçon trisectrix |
1/2 | Dürer folium |
2 | quadrifolium |
3 | trifolium |
The arc length of a single petal is
(3)
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where is the complete elliptic integral of the second kind, and the area of a petal is
(4)
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