Polar Graphs: Circles, Roses & Cardioids

You’ll be able to

Identify the polar curve family from the form of its equation
Predict the number of petals of a rose curve from its coefficient
Explain what a negative value of r means and how it is plotted

Families you can recognize on sight

A handful of forms cover nearly every polar curve on the exam. r = a is a circle centered at the origin. r = a·cos θ or a·sin θ is a circle of diameter a, passing through the origin, sitting along the horizontal or vertical axis. r = a·cos(nθ) is a rose. r = a ± b·sin θ with a = b is a cardioid, and with a ≠ b it is a limaçon. Recognizing the form saves plotting dozens of points.

Polar curve families

r = a: circle radius a · r = a cos θ: circle diameter a on the polar axis · r = a cos(nθ): rose · r = a ± b sin θ: limaçon, cardioid when a = b

r = a cos θ is a circle of *radius* a/2 centered at (a/2, 0), not a circle of radius a. The a is the diameter.

Counting rose petals

For r = a·cos(nθ) with n a positive integer, the petal count follows a rule that looks backward but is easy to remember: odd n gives n petals, even n gives 2n petals. So r = 3cos(3θ) has 3 petals and r = 3cos(2θ) has 4. The reason is that for odd n the second half of the revolution retraces the first half exactly, while for even n it produces a fresh set of petals in between. Each petal has length |a|.

Petal count and length

r = a·cos(nθ): n odd → n petals · n even → 2n petals · petal length |a|

A cosine rose has a petal along the polar axis; a sine rose is the same curve rotated. The count rule is identical for both.

Checkpoint

How many petals does r = 5 sin(4θ) have?

What a negative r means

The point (r, θ) with r < 0 is plotted by facing the direction θ and then walking backward — equivalently, it is the point (|r|, θ + π). So (−2, π/3) and (2, 4π/3) are the same point. Negative r is what allows a single continuous equation to trace petals on the opposite side of the origin, and it is why an odd rose retraces itself: on the second half-revolution r goes negative and lands back on the petals already drawn.

Worked example

Sketch r = 2 + 2 sin θ by finding r at the quadrantal angles, and identify the curve.

1.θ = 0: r = 2 + 0 = 2. The point is 2 units along the polar axis.
2.θ = π/2: r = 2 + 2 = 4. Maximum, straight up.
3.θ = π: r = 2 + 0 = 2. Two units to the left.
4.θ = 3π/2: r = 2 − 2 = 0. The curve reaches the origin, pointing down.
5.Since a = b = 2, this is a cardioid.

Answer: A cardioid opening upward, with maximum r = 4 at θ = π/2 and a cusp at the origin at θ = 3π/2. The cusp is the signature of a = b: the curve reaches r = 0 exactly once, and that is what distinguishes a cardioid from the limaçons on either side of it.

Tip

For any limaçon r = a ± b sin θ, evaluate at the four quadrantal angles 0, π/2, π and 3π/2. Those four values determine the shape — whether the curve has an inner loop (a < b), a cusp (a = b), or a dimple (a > b).

Converting between polar and rectangular

The conversions are x = r cos θ and y = r sin θ going one way, with r² = x² + y² and tan θ = y/x coming back. These turn recognition problems into algebra: multiplying r = 4 cos θ by r gives r² = 4r cos θ, which becomes x² + y² = 4x, then (x − 2)² + y² = 4 — visibly a circle of radius 2 centered at (2, 0). When a polar form is unfamiliar, converting often settles what the curve actually is.

Checkpoint

The polar equation r = 6 sin θ describes which curve?

Answer the 2 checkpoints as you read.