Rhodonea Curves

Rose (mathematics)

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).
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) = cosleft(k theta - frac{pi}{2} right) = cosleft(k left(theta-frac{pi}{2k} right) right)
for all theta, the curves given by the polar equations
,r=sin(ktheta) and ,r = cos(ktheta)
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.

Area

A rose whose polar equation is of the form
r=a cos (ktheta),
where k is a positive integer, has area
frac{1}{2}int_{0}^{2pi}(acos (ktheta))^2,dtheta = frac {a^2}{2} left(pi + frac{sin(4kpi)}{4k}right) = frac{pi a^2}{2} if k is even, and
frac{1}{2}int_{0}^{pi}(acos (ktheta))^2,dtheta = frac {a^2}{2} left(frac{pi}{2} + frac{sin(2kpi)}{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 (ktheta),
since the graphs of these are just rigid rotations of the roses defined using cosine.

See also

References

External links

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