Five petals from one formula. Three circles already make a recognisable flower.
Measuring the shape…
r = cos 5θ, 0 ≤ θ ≤ π
Rose curves (rhodonea) were named by Guido Grandi in the 1720s. With r = cos kθ, an odd k gives k petals and an even k gives 2k. A rose is really just two circles spinning in opposite directions at different speeds, so it needs very few to draw.
Each path of the rose curve is traced by a chain of circles, each one spinning on the tip of the one before. A Fourier transform measures the path and works out every circle's size, speed and starting angle. The circles are sorted biggest first, so the first few give the rough shape and the rest add detail.
"Within 1%" means the drawing is, on average, less than 1% of its width away from the real outline. The bars show the circles of the main outline: height is size, position is how many times it turns per loop and in which direction. Click a bar to remove that circle and see what it was doing. Learn how it works.