Rose curves: r = cos(kθ)
A rose curve, r = a cos(kθ), looks like a flower with petals arranged around the origin. The rule for counting the petals is simple and a little surprising: k petals when k is odd, 2k when k is even.
Equations to type
At a glance
- Petals
- k if k is odd, 2k if k is even
- Petal length
- |a|
- Total area
- πa²/4 for odd k, πa²/2 for even k
Why odd and even differ
As θ goes from 0 to 2π, cos(kθ) goes through k full cycles, each with a positive half and a negative half. Every half-cycle draws one petal, so 2k petals are drawn. When k is even, all 2k land in different places. When k is odd, each negative half draws its petal exactly on top of a positive one (because (−r, θ + π) is the same point as (r, θ)), so only k distinct petals appear.
Area of a rose
Each petal has area πa²/(4k). With 2k petals for even k the total is πa²/2; with k petals for odd k it is πa²/4. So a four-petal rose r = cos 2θ covers exactly half of the unit disc it sits in.
Fractional roses
When k = p/q is a fraction in lowest terms, the curve closes only after θ has gone round q or 2q times, and the petals overlap into intricate patterns. That is why the calculator draws polar curves over 0 ≤ θ ≤ 12π by default: it closes every rose with a denominator up to 6.
Frequently asked questions
How many petals does a rose curve have?
What is the difference between sin and cos roses?
How long are the petals?
Who studied rose curves?
More curves
r = 1 + cos θ: a circle rolling around a circle.
Several equations that draw a heart, and why they work.
The figure-eight curve and the infinity symbol.
Temple Fay’s transcendental butterfly from 1989.
(sin at, sin bt): patterns from two vibrations.
r = aθ, r = e^(bθ) and r² = θ: three kinds of spiral.
Every equation on this page opens in the calculator, where you can add sliders and watch the curve change.