Rose Curves r = a·cos(nθ)

Calculus 2 · Axiom Academy

LESSON Rose Curves r = a·cos(nθ) One polar equation, two parameters, and a flower — where the petal count is decided entirely by whether n is odd or even. A rose curve is defined by the polar equation below. The parameter a controls the size (the length of each petal), and n controls the number and arrangement of the petals. For each angle , the radius r is read off, then the point is plotted at . Watch a single petal-trio form for . The blue dot is the angle marching from 0 to ; the orange dot is the plotted point at radius , and the curve is its trail. The number of petals depends on n through a strikingly simple rule — and there are two cases, decided purely by the parity of n . If n is odd : the rose has exactly n petals. If n is even : the rose has 2n petals. Below, the two cases are drawn side by side from the same equation — left n=3 (odd), right n=4 (even). A counter ticks up by one every time a fresh petal tip is reached, so you can literally count along. While n fixes how many petals there are, a fixes how big they are. Changing a scales the whole rose without changing its petal count. Here three roses — all with n=5 — grow at . 4. Why Even n Doubles the Petals The whole rule comes down to one subtlety: what happens when r goes negative . When r < 0 , the point isn't dropped — it's plotted in the opposite direction, at angle . Here is n=2 traced slowly: the petals drawn while r > 0 are blue, and the petals drawn while r < 0 are red.

This is the written version of the interactive lesson above. See the full Calculus 2 course.