Rose (mathematics)

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Rose with k = 7 petals
Rose with k = 7 petals
Rose with 8 petals (k=4).
Rose with 8 petals (k=4).
Rose curves defined by r = sinkθ, for various values of k=n/d.
Rose curves defined by r = sinkθ, 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

\!\,r=\cos(k\theta).

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

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

for all θ, the curves given by the polar equations

\,r=\sin(k\theta) and \,r = \cos(k\theta)

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]

Contents

[edit] Area

A rose whose polar equation is of the form

r=a \cos (k\theta)\,

where k is a positive integer, has area

\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}

if k is even, and

\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}

if k is odd.

The same applies to roses with polar equations of the form

r=a \sin (k\theta)\,

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

[edit] See also

[edit] References

  1. ^ O'Connor, John J. & Robertson, Edmund F., “Rhodonea”, MacTutor History of Mathematics archive 

[edit] External links