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)
|
(e.g., Lawrence 1972, p. 175; Ferréol; illustrated above) or by the version rotated by 
,
 |
(2)
|
(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)
|
where 
is the complete
elliptic integral of the second kind, and the area of a petal is
 |
(4)
|
