The Rose Curve

May 22, 2026

A rose curve is a polar curve that produces a pattern of evenly spaced petals. The equation is deceptively simple:

$$r = \cos(n\theta)$$

But the behavior is surprising: depending on whether $n$ is odd or even, the number of petals changes dramatically.

Polar Coordinates, Briefly

Before diving in, recall how polar coordinates work. Instead of describing a point by its horizontal and vertical distances $(x, y)$, polar coordinates use:

$r$ — the distance from the origin
$\theta$ — the angle measured counterclockwise from the positive $x$-axis

A polar curve is defined by specifying $r$ as a function of $\theta$. As $\theta$ sweeps from $0$ to $2\pi$, the point $(r, \theta)$ traces a path in the plane.

The circle $r = 5$ is the simplest example — the radius is constant, so every angle gives a point at the same distance from the origin. The rose curve is more interesting: the radius oscillates as $\theta$ increases, which is what creates the petals.

The Equation

The standard rose curve is:

$$r = a\cos(n\theta) \qquad \text{or} \qquad r = a\sin(n\theta)$$

where $a > 0$ controls the length of each petal and $n$ is a positive integer that controls the number of petals.

The sine and cosine versions produce the same shape, just rotated by $\frac{\pi}{2n}$ radians. This is because $\sin(\theta) = \cos(\theta - \pi/2)$, so the sine version is simply the cosine version with a phase shift. For simplicity, we'll focus on $r = \cos(n\theta)$.

Counting the Petals

The most striking property of the rose curve is the petal counting rule:

If $n$ is odd, the curve has exactly $n$ petals.
If $n$ is even, the curve has exactly $2n$ petals.

This asymmetry is not obvious from the equation. Let's look at some examples.

3 Petals: $r = \cos(3\theta)$

With $n = 3$ (odd), we get 3 petals. As $\theta$ runs from $0$ to $\pi$, the function $\cos(3\theta)$ completes $\frac{3}{2}$ full cycles. Each positive lobe of $\cos(3\theta)$ traces one petal. The remaining half of the journey, from $\theta = \pi$ to $2\pi$, retraces the same three petals — so the curve closes after one full revolution with exactly 3 distinct petals.

4 Petals: $r = \cos(2\theta)$

With $n = 2$ (even), we get 4 petals — double what you might expect. As $\theta$ runs from $0$ to $\pi$, $\cos(2\theta)$ completes a full cycle and traces 2 petals. Then from $\pi$ to $2\pi$, it traces 2 more petals in different directions rather than retracing the first two. This is because when $n$ is even, replacing $\theta$ with $\theta + \pi$ produces the same $r$ value:

$$\cos!\bigl(n(\theta + \pi)\bigr) = \cos(n\theta + n\pi) = \cos(n\theta)\cos(n\pi) - \sin(n\theta)\sin(n\pi)$$

When $n$ is even, $\cos(n\pi) = 1$ and $\sin(n\pi) = 0$, so the expression equals $\cos(n\theta)$. The second half of the revolution lands on different points rather than repeating the first half — giving twice as many petals.

When $n$ is odd, $\cos(n\pi) = -1$, so $r$ flips sign. A negative $r$ in polar coordinates means "go in the opposite direction," which sends the point to the same petal already traced.

8 Petals: $r = \cos(4\theta)$

With $n = 4$ (even), we get $2 \times 4 = 8$ petals. Notice how the petals are arranged symmetrically around the origin, alternating between the axes and the diagonals.

Why the Petals Form

Each petal corresponds to a single positive arch of $\cos(n\theta)$. The function $\cos(n\theta)$ is positive when $n\theta \in (-\pi/2, \pi/2)$, i.e., when $\theta \in (-\frac{\pi}{2n}, \frac{\pi}{2n})$. On this interval, $r$ grows from $0$ to $a$ and back to $0$, tracing one complete petal.

The center of each petal lies along the direction $\theta = \frac{k\pi}{n}$ for integer $k$. The petal tip is at distance $a$ from the origin, and the petal tapers to a point at the origin at each end.

Crucially, a petal only appears when $r \geq 0$. When $\cos(n\theta) < 0$, polar convention sends the point in the opposite direction — for odd $n$, this retraces an existing petal; for even $n$, it creates a new one in a fresh direction.

The Name

The rose curve was named rhodonea (from the Greek $\rho\acute{o}\delta o\nu$, rose) by the Italian mathematician Guido Grandi around 1728. Grandi studied these curves extensively and was struck by their resemblance to flowers — the same intuition that makes them one of the most visually appealing objects in elementary mathematics.

Conclusion

The rose curve $r = \cos(n\theta)$ is a perfect example of how a small change in a parameter can have a dramatic structural effect. The odd/even rule for petal count is not immediately obvious from the equation but falls out naturally from the sign behavior of cosine — and from the way polar coordinates handle negative radii. That a single integer $n$ can produce such a variety of symmetric, flower-like shapes is one of the quiet pleasures of polar geometry.