Rose Curve
Polar curvesThe polar curve r = a·cos(kθ) that blooms into k or 2k petals depending on whether k is odd or even.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
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.
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.
Where you'll see it
Rose curves appear in antenna radiation patterns, vibration mode diagrams, and virtually every polar-coordinates lesson.