Rose (mathematics)

Rose with 8 petals (k=4).

Rose curves defined by <math>r=\sin k \theta</math>, for various values of k=n/d.

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

<math>\!\,r=\cos(k\theta).</math>

If k is an integer, the curve will be rose shaped with

• 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

<math>\sin(k \theta) = \cos\left( k \theta - \frac{\pi}{2} \right) = \cos\left( k \left( \theta-\frac{\pi}{2k} \right) \right)</math>

for all <math>\theta</math>, the curves given by the polar equations

<math>\,r=\sin(k\theta)</math> and <math>\,r = \cos(k\theta)</math>

are identical except for a rotation of π/2k radians. Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728.^[1]

Area

A rose whose polar equation is of the form

<math>r=a \cos (k\theta)\,</math>

where k is a positive integer, has area

<math>

   \frac{1}{2}\int_{0}^{2\pi}(a\cos (k\theta))^2\,d\theta = \frac {a^2}{2} \left(\pi + \frac{\sin(4k\pi)}{4k}\right) = \frac{\pi a^2}{2}

</math> if k is even, and

<math>

   \frac{1}{2}\int_{0}^{\pi}(a\cos (k\theta))^2\,d\theta = \frac {a^2}{2} \left(\frac{\pi}{2} + \frac{\sin(2k\pi)}{4k}\right) = \frac{\pi a^2}{4}

</math> if k is odd. The same applies to roses with polar equations of the form

<math>r=a \sin (k\theta)\,</math>

since the graphs of these are just rigid rotations of the roses defined using cosine.

See also

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

References

1. ^ O'Connor, John J; Edmund F. Robertson "Rhodonea". MacTutor History of Mathematics archive.  

External links

• Eric W. Weisstein, Rose at MathWorld.
• Applet to create rose with k parameter