# Rose (mathematics)

In mathematics, a **rose** (also called a rhodonea, Grandi's rose, or multifolium) is a plane curve whose equation in polar coordinates has the form \( r = a \sin(k\theta) \) or \( r = a \cos(k\theta) \), where \( a \) and \( k \) are constants; the curve traces petal-shaped loops resembling a flower. [The Italian](https://www.edgepedia.org/the-italian) mathematician Guido Grandi, a Camaldolese who became professor of mathematics at the University of Pisa in 1714, named the curves *rhodonea* between 1723 and 1728 because of this resemblance, and he was the first to describe them, in 1728.<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Rhodonea/)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup><sup> • </sup><sup>[3](https://mathworld.wolfram.com/RoseCurve.html)</sup>

| Key fact | Detail |
|---|---|
| Defining equation | \( r = a \sin(k\theta) \) or \( r = a \cos(k\theta) \), with \( a \neq 0 \)<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Rhodonea/)</sup><sup> • </sup><sup>[4](https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_%28OpenStax%29/10%3A_Further_Applications_of_Trigonometry/10.04%3A_Polar_Coordinates_-_Graphs)</sup> |
| Petal count | For integer \( k \): \( k \) petals if \( k \) is odd, \( 2k \) petals if \( k \) is even<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup> |
| Rational \( k = m/n \) | An algebraic curve of even order, of order \( m+n \) if \( m \) and \( n \) are both odd and \( 2(m+n) \) if either is even<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup> |
| Irrational \( k \) | Infinitely many petals<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup> |
| Size | The whole curve lies inside the circle of radius \( a \)<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup> |
| Area of one petal | \( S = \pi a^{2}/(4k) \)<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup> |
| Arc length | Given by an elliptic integral of the second kind<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup> |

## How the parameter k shapes the curve

The number of petals depends on \( k \) in a way that surprises many students. When \( k \) is an odd integer, only \( k \) petals appear, because the negative \( r \)-values retrace petals already drawn; when \( k \) is even, the \( 2k \) petals are all distinct. For example, \( r = 2\cos(4\theta) \) produces eight petals, each reaching a maximum radius of 2 and returning to the pole.<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup><sup> • </sup><sup>[4](https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_%28OpenStax%29/10%3A_Further_Applications_of_Trigonometry/10.04%3A_Polar_Coordinates_-_Graphs)</sup> When \( k \) is irrational the curve never closes and fills the disk with infinitely many petals.<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup>

For a rational \( k = m/n \) in lowest terms, the rose consists of \( m \) petals when \( m \) and \( n \) are both odd, and \( 2m \) petals when either is even.<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup> In this case the curve is algebraic, with the degree stated in the table above.<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup>

The choice between sine and cosine only rotates the figure: the sine form gives a petal oriented vertically for odd \( k \), while the cosine form gives a petal oriented to the right.<sup>[3](https://mathworld.wolfram.com/RoseCurve.html)</sup>

## Special cases and related curves

The **quadrifolium** is the rose with \( k = 2 \), with polar equation \( r = a\sin(2\theta) \) and Cartesian form \( (x^{2}+y^{2})^{3} = 4a^{2}x^{2}y^{2} \).<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Rhodonea/)</sup> Roses belong to the family of cycloidal curves: they are hypocycloids when \( k > 1 \) and epicycloids when \( k < 1 \), and they are also the pedals of epi- and hypocycloids with respect to the center of the fixed circle.<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup> They can also be regarded as polar variants of bivariate Lissajous curves.<sup>[5](https://www.math.unipd.it/~erb/rhodonea.html)</sup>

## Measurement

The area of one petal is \( S = \pi a^{2}/(4k) \), so the total area traced by the curve is a fixed fraction of the circumscribed circle's area: half when \( k \) is an even integer and a quarter when \( k \) is odd.<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup><sup> • </sup><sup>[6](https://mathcurve.com/courbes2d.gb/rosace/rosace.shtml)</sup> The arc length of a rose is given by an elliptic integral of the second kind, so it has no simpler closed form in general.<sup>[2](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)</sup>

## Modern uses

Rose curves have found a role in numerical analysis. Wolfgang Erb, a mathematician at the [University of Padua](https://www.edgepedia.org/university-of-padua), showed in 2018 that point samples along rhodonea curves can serve as node sets for spectral interpolation on the unit disk, using a parity-modified Chebyshev-Fourier basis; for rectangular spectral index sets the interpolant is continuous at the center, the Lebesgue constant grows logarithmically, and the scheme converges quickly for smooth functions.<sup>[5](https://www.math.unipd.it/~erb/rhodonea.html)</sup><sup> • </sup><sup>[7](https://arxiv.org/abs/1812.00437)</sup> The same nodes support a Clenshaw-Curtis quadrature rule, and the scheme has been demonstrated by interpolating image data of the [black hole shadow](https://www.edgepedia.org/black-hole-shadow) at the center of galaxy M87 from the [Event Horizon Telescope](https://www.edgepedia.org/event-horizon-telescope) collaboration.<sup>[7](https://arxiv.org/abs/1812.00437)</sup><sup> • </sup><sup>[8](https://www.math.unipd.it/~erb/pdf/poster-iciam.pdf)</sup>

In teaching, rose curves are a standard example in polar-coordinate graphing because a single simple equation generates an intricate-looking figure, and sketching proceeds by locating the zeros of \( r \) and the petal tips.<sup>[4](https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_%28OpenStax%29/10%3A_Further_Applications_of_Trigonometry/10.04%3A_Polar_Coordinates_-_Graphs)</sup>

## References

1. [Rhodonea Curves, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Curves/Rhodonea/)
2. [Roses (curves), Encyclopedia of Mathematics (D.D. Sokolov)](https://encyclopediaofmath.org/wiki/Roses_%28curves%29)
3. [Rose Curve, Wolfram MathWorld](https://mathworld.wolfram.com/RoseCurve.html)
4. [Polar Coordinates - Graphs, Mathematics LibreTexts (OpenStax)](https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_%28OpenStax%29/10%3A_Further_Applications_of_Trigonometry/10.04%3A_Polar_Coordinates_-_Graphs)
5. [Wolfgang Erb, Rhodonea curves, University of Padua](https://www.math.unipd.it/~erb/rhodonea.html)
6. [Rose, mathcurve.com](https://mathcurve.com/courbes2d.gb/rosace/rosace.shtml)
7. [W. Erb (2018), Rhodonea curves as sampling trajectories for spectral interpolation on the unit disk, arXiv:1812.00437](https://arxiv.org/abs/1812.00437)
8. [W. Erb, Spectral Interpolation Schemes on Curves in Polar and Spherical Geometries, ICIAM 2019 poster](https://www.math.unipd.it/~erb/pdf/poster-iciam.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Analytic and coordinate geometry*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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