Polar Roses

Polar rose curves, also called rhodonea curves and studied by the Italian mathematician Luigi Guido Grandi in the early 18th century, are produced by polar equations of the form r = \sin(n\theta) or r = \cos(n\theta). The integer n controls petal count and arrangement: odd n yields n petals, while even n yields 2n petals. Roses are a classic example of how polar coordinates make some shapes far more compact to describe than equivalent Cartesian equations would.

Here a single rose, (\sin(k\,t)\cos t, \sin(k\,t)\sin t), is drawn over one full turn while the integer k steps through \{2, 3, \dots, 8\}. Following the rule above — odd k gives k petals, even k gives 2k — the petal count visibly jumps at each step, a clean demonstration of a discrete, stepped parameter rather than a continuous sweep. The equal-aspect viewport keeps the petals undistorted.