Worked example
Polar Rose: r = 2·cos(3θ)
In polar coordinates, each point is described by a distance r from the origin and an angle θ, and the equation r = 2·cos(3θ) draws a flower with exactly three petals. As θ sweeps around, r oscillates between -2 and 2 three times; negative r values plot in the opposite direction, which is what folds the petals into their symmetric arrangement.
The number of petals follows a simple rule: for r = a·cos(nθ) with odd n, the rose has exactly n petals. Try even values of n in the calculator and you will get twice as many — the pattern doubles because the curve needs a full extra revolution to close. Few equations show off the power of polar coordinates as elegantly as the rose.
Plotted expressions
- r = 2*cos(3*theta)
The link preloads these exact expressions into the calculator — no typing needed.
What to notice
- An odd coefficient (3) gives 3 petals; try 2*cos(4*theta) to see the even case produce 8.
- The amplitude 2 sets the petal length — the farthest point from the origin.
- Each petal is traced exactly once as θ runs from 0 to π; the second half retraces them.