Polar Curves: Rose, Cardioid, and Limaçon

Polar curves produce some of the most beautiful shapes in mathematics. By expressing the radius as a function of angle, $r = f(\theta)$, we get curves that would be extremely complicated in Cartesian form.

Rose curves: $r = \cos(n\theta)$

• Odd $n$: $n$ petals (e.g., $n = 3$ gives 3 petals)
• Even $n$: $2n$ petals (e.g., $n = 2$ gives 4 petals)
• Each petal has length 1 (the coefficient of cosine)

Cardioid: $r = 1 + \cos\theta$

Heart-shaped curve. At $\theta = 0$, $r = 2$ (farthest from origin). At $\theta = \pi$, $r = 0$ (passes through the origin). The name comes from Greek "kardia" (heart).

Limaçon: $r = a + b\cos\theta$

• If $a > b$: convex limaçon (no loop, no dimple)
• If $a = b$: cardioid (special case)
• If $a < b$: limaçon with inner loop (the curve crosses the origin)

For $r = 1 + 2\cos\theta$: $a = 1$, $b = 2$, so $a < b$ → inner loop.

Try it in the visualization

Select a curve type and watch the point trace the curve as θ sweeps from 0 to 2π. The rose curve slider lets you change $n$ to see how the petal count changes. Toggle between different limaçon shapes by adjusting $a/b$.