Polar Coordinates and Graphs

Introduction to Polar Coordinates

In the polar coordinate system, each point is determined by:

• $r$ = distance from the origin (pole)
• $\theta$ = angle from the positive x-axis (polar axis)

A point is written as $(r, \theta)$ in polar form.

Conventions

• $r$ can be positive or negative
  • If $r > 0$, move $r$ units in the direction of $\theta$
  • If $r < 0$, move $|r|$ units in the opposite direction of $\theta$
• $\theta$ is typically measured in radians
• Multiple representations: $(r, \theta)$, $(r, \theta + 2\pi n)$, $(-r, \theta + \pi)$ all represent the same point

Conversion Formulas

Polar to Rectangular

Given $(r, \theta)$ in polar: $x = r\cos(\theta)$ $y = r\sin(\theta)$

Rectangular to Polar

Given $(x, y)$ in rectangular: $r = \sqrt{x^2 + y^2}$ $\theta = \arctan(\frac{y}{x})$ (adjust for quadrant)

Or use: $\tan(\theta) = \frac{y}{x}$

Important: When finding $\theta$:

• Use $\arctan(\frac{y}{x})$ as starting point
• Adjust based on the quadrant of $(x, y)$
• Quadrant I: $\theta = \arctan(\frac{y}{x})$
• Quadrant II: $\theta = \pi + \arctan(\frac{y}{x})$
• Quadrant III: $\theta = \pi + \arctan(\frac{y}{x})$
• Quadrant IV: $\theta = 2\pi + \arctan(\frac{y}{x})$ or $\theta = \arctan(\frac{y}{x})$ (negative)

Common Polar Graphs

Circles

• $r = a$: Circle centered at origin with radius $a$
• $r = 2a\cos(\theta)$: Circle with diameter $2a$ on the polar axis
• $r = 2a\sin(\theta)$: Circle with diameter $2a$ perpendicular to polar axis

Lines

• $\theta = \alpha$: Line through origin at angle $\alpha$
• $r\cos(\theta) = a$: Vertical line $x = a$
• $r\sin(\theta) = a$: Horizontal line $y = a$

Limaçons

General form: $r = a \pm b\cos(\theta)$ or $r = a \pm b\sin(\theta)$

• If $\frac{a}{b} < 1$: Inner loop
• If $\frac{a}{b} = 1$: Cardioid (heart-shaped)
• If $1 < \frac{a}{b} < 2$: Dimpled limaçon
• If $\frac{a}{b} \geq 2$: Convex limaçon

Rose Curves

General form: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$

• If $n$ is odd: $n$ petals
• If $n$ is even: $2n$ petals
• Length of each petal: $|a|$

Lemniscates

• $r^2 = a^2\cos(2\theta)$: Figure-eight along polar axis
• $r^2 = a^2\sin(2\theta)$: Figure-eight at $45°$

Spirals

• $r = a\theta$: Archimedean spiral
• $r = ae^{b\theta}$: Exponential spiral

Symmetry in Polar Graphs

Test for symmetry to help sketch graphs:

1. Symmetry about the polar axis (x-axis):

   • Replace $(r, \theta)$ with $(r, -\theta)$
   • If equation unchanged, symmetric about polar axis

2. Symmetry about $\theta = \frac{\pi}{2}$ (y-axis):

   • Replace $(r, \theta)$ with $(r, \pi - \theta)$ or $(-r, -\theta)$
   • If equation unchanged, symmetric about $\theta = \frac{\pi}{2}$

3. Symmetry about the pole (origin):

   • Replace $(r, \theta)$ with $(-r, \theta)$ or $(r, \theta + \pi)$
   • If equation unchanged, symmetric about the pole

Graphing Strategy

1. Identify the type of polar curve
2. Check for symmetry
3. Create a table of values for $\theta$ from $0$ to $2\pi$
4. Plot key points and note special values
5. Sketch the curve connecting points smoothly
6. Consider domain restrictions if $r^2$ appears (must have $r^2 \geq 0$)