Understanding Polar Curves

A polar curve is a shape constructed using the polar coordinate system, where each point on the graph is determined by a distance from the origin $r$ and an angle from the positive x-axis $\theta$. These curves are typically defined by equations of the form $r = f(\theta)$.

Common Types of Polar Curves

Understanding the standard formulas helps you quickly identify the shape of the graph before plotting any points.

1. Circles

• Centered at the pole: $r = a$
• Touching the pole: $r = a\cos\theta$ (lies along the x-axis) or $r = a\sin\theta$ (lies along the y-axis).

2. Limaçons and Cardioids

These follow the form $r = a \pm b\cos\theta$ or $r = a \pm b\sin\theta$, where $a > 0$ and $b > 0$.

• Cardioid: When $a = b$, the curve is heart-shaped and has a sharp point (cusp) at the pole.
• Limaçon with an inner loop: When $a < b$.
• Dimpled / Convex Limaçon: When $a > b$.

3. Rose Curves

Equations of the form $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$.

• The curve features "petals" of maximum length $a$.
• If $n$ is odd, the rose has exactly $n$ petals.
• If $n$ is even, the rose has $2n$ petals.

Symmetry Tests for Graphing

Testing for symmetry allows you to sketch curves by only calculating points for a fraction of the graph:

• Symmetry about the Polar Axis (x-axis): The equation remains unchanged when $\theta$ is replaced by $-\theta$. (Common with cosine functions).
• Symmetry about $\theta = \pi/2$ (y-axis): The equation remains unchanged when $\theta$ is replaced by $\pi - \theta$. (Common with sine functions).
• Symmetry about the Pole (origin): The equation remains unchanged when $r$ is replaced by $-r$ or $\theta$ by $\theta + \pi$.

Example Problems

Example 1: Graph $r = 3\cos(2\theta)$ and identify the curve type

Analysis: This matches the form $r = a\cos(n\theta)$, which is a rose curve.

• The amplitude $a = 3$ means the maximum length of each petal is $3$.
• The coefficient $n = 2$. Since $2$ is an even number, the curve will have $2n = 2(2) = 4$ petals.
• Because it uses cosine, the first petal is centered directly on the polar axis ($\theta = 0$). The four petals are spaced evenly every $\pi/2$ radians ($90^\circ$).

Example 2: Graph $r = 2 + 2\sin\theta$

Analysis: This matches the form $r = a + b\sin\theta$.

• Because $a = 2$ and $b = 2$, we have $a = b$, making this specific limaçon a cardioid.
• Since the equation uses $+ \sin\theta$, the "fat" part of the heart is oriented upwards along the positive y-axis ($\theta = \pi/2$).
• Key points:
  • At $\theta = 0$, $r = 2 + 0 = 2$.
  • At $\theta = \pi/2$, $r = 2 + 2(1) = 4$ (the peak of the curve).
  • At $\theta = \pi$, $r = 2 + 0 = 2$.
  • At $\theta = 3\pi/2$, $r = 2 + 2(-1) = 0$ (the curve hits the pole, forming the cusp).