Shapes on Polar Coordinate Systems

When graphing on a polar graph, you can generate some very interesting shapes. Some of these include limaçons, rose curves, circles, and lemniscates.

Limaçons:

r = a ± b cosθ
r = a ± b sinθ
(0 < a, 0 < b)

Rose curves (if n is odd, there will be n "petals"; if n is even, there will be 2n "petals"):

r = a cos(nθ)
r = a sin(nθ)
(n ≥ 2)

Circles:

r = a cosθ
r = a sinθ

Lemniscates:

r² = a² sin(2θ)
r² = a² cos(2θ)

Here are some specific pictures (from our book) of these graphs: