Rose Curve
The polar curve r = a·cos(kθ) that blooms into k or 2k petals depending on whether k is odd or even.
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Formulas and key values
- Polar equation
- Petals · odd k
- Petals · even k
- Area · odd k
- Area · even k
a is the petal length (a = 1 in the graph) and k decides the number of petals.
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Start the video01 · Definition
What is the Rose Curve?
The rose curve, or rhodonea, is the polar graph r = a·cos(kθ). As θ sweeps around the origin, the radius oscillates between a and −a, folding the circle into evenly spaced petals. The curve was named and studied by the Italian mathematician Guido Grandi in the 1720s.
The petal count follows a simple parity rule: when k is odd the curve closes after half a turn and shows exactly k petals; when k is even it needs a full turn and shows 2k petals. Watching the count jump as k moves from 4 to 5 is one of the clearest demonstrations of how polar symmetry works.
02 · Properties
Key properties
- k odd → k petals traced in θ ∈ [0, π); k even → 2k petals over a full turn.
- Each petal spans an angle of π/k and has maximum radius a.
- The total enclosed area is πa²/4 for odd k and πa²/2 for even k.
- Using sin(kθ) instead of cos(kθ) rotates the whole rose by π/(2k).
- Non-integer k produces overlapping petals that may never close.
03 · Questions
Rose Curve: common questions
- What is the equation of the Rose Curve?
- Polar equation: r = a cos(kθ). a is the petal length (a = 1 in the graph) and k decides the number of petals.
- What is the area of the Rose Curve?
- Area (odd k): A = (π a^2)/4; Area (even k): A = (π a^2)/2. a is the petal length (a = 1 in the graph) and k decides the number of petals.
04 · Keep exploring