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Pre-Calculus Parametric and polar

Polar graphs (rose curves, cardioids, limaçons)

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Theory

A polar graph plots \(r=f(\theta)\): for each angle \(\theta\), the radius \(f(\theta)\) sets how far out the point lies. Several famous families appear:

  • Circle: \(r=a\) (radius \(a\)), or \(r=a\cos\theta\)/\(a\sin\theta\).
  • Rose curve: \(r=a\cos(n\theta)\) or \(a\sin(n\theta)\) — \(n\) petals if \(n\) is odd, \(2n\) if even.
  • Cardioid: \(r=a\pm a\cos\theta\) — a heart shape.
  • Limaçon: \(r=a\pm b\cos\theta\) — with an inner loop when \(\dfrac{a}{b}<1\).
Recognize the family from the form, then use symmetry and a few key angles to sketch.
Rose curve A three-petaled rose r equals 3 cosine 3 theta.
A rose curve \(r=3\cos(3\theta)\) — three petals.
Cardioid A cardioid r equals 2 times one plus cosine theta.
A cardioid \(r=2(1+\cos\theta)\).

The main polar families:

\[r=a\ (\text{circle}),\quad r=a\cos(n\theta)\ (\text{rose}),\quad r=a\pm b\cos\theta\ (\text{lima\c{c}on})\]
polar circles, rose curves, and limacons including the cardioid
Rose petals: \(n\) petals if \(n\) is odd, \(2n\) if even.

How to graph a polar equation

  1. Recognize the family from the form of \(r=f(\theta)\).
  2. Use symmetry (about the axes or pole) to reduce work.
  3. Plot key angles \(0,\dfrac{\pi}{2},\pi,\dfrac{3\pi}{2}\) and any where \(r=0\).
  4. Connect smoothly as \(\theta\) sweeps around.
Example 1 — A circle
Describe the polar graph \(r=4\).
Solution

Every point is \(4\) units from the pole, at any angle.

r=4\(\Rightarrow\)\text{circle, radius }4
r equals 4 is a circle of radius 4
Example 2 — A rose curve
How many petals does \(r=3\cos(3\theta)\) have?
Solution

For \(r=a\cos(n\theta)\), an odd \(n\) gives \(n\) petals.

n=3\ (\text{odd})\(\Rightarrow\)3\ \text{petals}
three petals
Example 3 — A cardioid
What shape is \(r=2+2\cos\theta\)?
Solution

A limaçon with \(a=b\) is a cardioid (heart shape).

a=b=2\(\Rightarrow\)\text{cardioid}
it is a cardioid
Example 4 — Limaçon type
Classify \(r=1+3\cos\theta\).
Solution

For \(r=a+b\cos\theta\), \(\dfrac{a}{b}<1\) gives an inner-loop limaçon.

\dfrac{a}{b}=\dfrac{1}{3}<1\(\Rightarrow\)\text{inner loop}
a limaçon with an inner loop

Common pitfalls

Rose petal count depends on parity. Odd \(n\) gives \(n\) petals; even \(n\) gives \(2n\).
Cardioid vs limaçon. \(a=b\) is a cardioid; \(\dfrac{a}{b}<1\) adds an inner loop.
\(r\) can be negative, which places the point across the pole.

Frequently asked questions

What is a polar graph?

The graph of \(r=f(\theta)\), where the radius depends on the angle, producing curves like circles, roses, and cardioids.

How many petals does a rose curve have?

For \(r=a\cos(n\theta)\): \(n\) petals when \(n\) is odd, and \(2n\) when \(n\) is even.

What is a cardioid?

A heart-shaped limaçon \(r=a\pm a\cos\theta\), where the constant and coefficient are equal.

When does a limaçon have an inner loop?

When \(\dfrac{a}{b}<1\) in \(r=a\pm b\cos\theta\).

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