Rose (mathematics)
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In mathematics, a rose is a sinusoid plotted in polar coordinates. Up to similarity,
- <math>\!\,r=cos(k\theta)<math>
One obtains a rose-like graph with <math>2k<math> petals if <math>k<math> is even and <math>k<math> petals if <math>k<math> is odd. Assuming you use the given form, the whole rose will appear inside a unit circle. Using sine instead of cosine, and vice versa, the graphs differ by a rotation of <math>\frac{\pi}{2}<math> radians—or that <math>\sin(kt + \frac{\pi}{2}) = \cos(kt)<math>, and the graphs coincide.
More interesting results arise when <math>k<math> is a rational. If <math>k<math> is irrational, without bounds on <math>\,\!\theta<math>, a disc results. In more detail, if <math>k<math> is irrational, the number of petals is irrational, and the only thing preventing you from a solid-appearing disc is the upper limit on <math>\,\!\theta<math>. Assuming a <math>k<math> of <math>\pi<math>, a <math>\,\!\theta<math> limit of 2520 degrees (14<math>\pi<math> radians) will give you the first complete circle.
External links
- Mathworld article on rose curves (http://mathworld.wolfram.com/Rose.html)Template:Math-stub