Rose (mathematics)

thumb|200px|right|Rose curves defined by ">$r=sin\; k\; heta$, for various values of k=n/d.

In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. Up to similarity, thesecurves can all be expressed by a polar equation of the form:$!,r=cos(k\; heta).$If "k" is an integer, the curve will be rose shaped with
*2"k" 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:$sin(k\; heta)\; =\; cosleft(\; k\; heta\; -\; frac\{pi\}\{2\}\; ight)\; =\; cosleft(\; k\; left(\; heta-frac\{pi\}\{2k\}\; ight)\; ight)$ for all $heta$, the curves given by the polar equations :$,r=sin(k\; heta)$ and $,r\; =\; cos(k\; heta)$are identical except for a rotation of π/2"k" radians.

Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728. [MacTutor Biography|class=Curves|id=Rhodonea|title=Rhodonea]

Area

A rose whose polar equation is of the form:$r=a\; cos\; (k\; heta),$where "k" is a positive integer, has area:$frac\{1\}\{2\}int\_\{0\}^\{2pi\}(acos\; (k\; heta))^2,d\; heta\; =\; frac\; \{a^2\}\{2\}\; left(pi\; +\; frac\{sin(4kpi)\}\{4k\}\; ight)\; =\; frac\{pi\; a^2\}\{2\}$if "k" is even, and:$frac\{1\}\{2\}int\_\{0\}^\{pi\}(acos\; (k\; heta))^2,d\; heta\; =\; frac\; \{a^2\}\{2\}\; left(frac\{pi\}\{2\}\; +\; frac\{sin(2kpi)\}\{4k\}\; ight)\; =\; frac\{pi\; a^2\}\{4\}$if "k" is odd.

The same applies to roses with polar equations of the form:$r=a\; sin\; (k\; heta),$since the graphs of these are just rigid rotations of the roses defined using cosine.

ee also

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

References

External links

*mathworld|urlname=Rose|title=Rose
* [http://www25.brinkster.com/denshade/PolarFlower.html Applet to create rose with k parameter]
* [http://xahlee.org/SpecialPlaneCurves_dir/Rose_dir/rose.html Visual Dictionary of Special Plane Curves] Xah Lee