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In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. Up to similarity, these curves can all be expressed by a polar equation of the form

- $!,r=cos(ktheta).$

- 2k petals if k is even, and
- k petals if k is odd.

When k is even, the entire graph of the rose will be traced out exactly once when the value of θ changes from 0 to 2π. When k is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for k even, and π for k odd.)

If k is rational, then the curve is closed and has finite length. If k is irrational, then it is not closed and has infinite length. Furthermore, the graph of the rose in this case forms a dense set (i.e., it comes arbitrarily close to every point in the unit disk).

Since

- $sin(k\; theta)\; =\; cosleft(k\; theta\; -\; frac\{pi\}\{2\}\; right)\; =\; cosleft(k\; left(theta-frac\{pi\}\{2k\}\; right)\; right)$

- $,r=sin(ktheta)$ and $,r\; =\; cos(ktheta)$

Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728.

- $r=a\; cos\; (ktheta),$

- $$

- $$

The same applies to roses with polar equations of the form

- $r=a\; sin\; (ktheta),$

- Lissajous curve
- quadrifolium - a rose curve with k=2.

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Last updated on Wednesday September 24, 2008 at 05:45:16 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday September 24, 2008 at 05:45:16 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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