10.4 Polar Coordinates: Graphs

Graphing Polar Equations

Introduction to Polar Coordinates

In polar coordinates, you describe a point by how far it is from the origin and what angle it makes, rather than using x and y. This system makes certain curves much easier to express and graph.

Every point on the polar plane has two components:

• Radial distance ($r$): how far the point is from the origin (called the pole)
• Angular coordinate ($\theta$): the angle measured counterclockwise from the positive x-axis (called the polar axis)

So the point $(3, \frac{\pi}{4})$ sits 3 units from the origin along a 45° angle.

Graphing Polar Equations

To graph a polar equation $r = f(\theta)$, follow these steps:

1. Build a table of values. Choose $\theta$ values from $0$ to $2\pi$, usually in increments of $\frac{\pi}{6}$ (30°) or $\frac{\pi}{4}$ (45°). Plug each $\theta$ into the equation to calculate the corresponding $r$.
2. Plot the points $(r, \theta)$ on a polar grid. The angle $\theta$ tells you the direction from the pole (0° points to the right, $\frac{\pi}{2}$ points straight up). The value $r$ tells you how far to go in that direction.
3. Connect the points smoothly to form the polar curve. Don't use straight line segments; these curves are typically rounded.

What about negative $r$? If your equation gives a negative $r$ for some $\theta$, you plot that point in the opposite direction. For example, $(-2, \frac{\pi}{3})$ is plotted 2 units from the pole at angle $\frac{\pi}{3} + \pi = \frac{4\pi}{3}$.

Symmetry shortcuts can save you a lot of work:

• Symmetric about the polar axis (x-axis) if $f(-\theta) = f(\theta)$
• Symmetric about the pole (origin) if $f(\theta + \pi) = f(\theta)$
• Symmetric about the line $\theta = \frac{\pi}{2}$ (y-axis) if $f(-\theta) = -f(\theta)$

If you confirm one of these, you only need to plot half (or less) of the curve and reflect.

Common Polar Curve Types

Cardioids have the form $r = a(1 \pm \cos\theta)$ or $r = a(1 \pm \sin\theta)$.

The graph is a heart-shaped curve that passes through the pole exactly once. For example, $r = 3(1 + \cos\theta)$ produces a cardioid that extends 6 units to the right and touches the pole on the left. The $\cos$ versions are symmetric about the polar axis; the $\sin$ versions are symmetric about the line $\theta = \frac{\pi}{2}$. The sign inside the parentheses determines which direction the curve "points": a plus sign with cosine means the cusp (the pointy part at the pole) faces left.

Limaçons have the form $r = a \pm b\cos\theta$ or $r = a \pm b\sin\theta$, where $a, b > 0$.

The shape depends on the ratio $\frac{a}{b}$:

• $\frac{a}{b} < 1$: the limaçon has an inner loop (the curve crosses through the pole and loops back on itself)
• $\frac{a}{b} = 1$: you get a cardioid (special case)
• $1 < \frac{a}{b} < 2$: the limaçon has a dimple but no inner loop
• $\frac{a}{b} \geq 2$: the limaçon is convex (no dimple, roughly egg-shaped)

Notice the original guide had the inner loop and dimple conditions swapped. The inner loop appears when $a < b$, not when $a > b$.

Rose curves have the form $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$, where $n$ is a positive integer.

• If $n$ is odd, the curve has exactly $n$ petals (e.g., $n = 3$ gives 3 petals)
• If $n$ is even, the curve has $2n$ petals (e.g., $n = 4$ gives 8 petals)
• The length of each petal equals $|a|$, so $r = 5\cos(3\theta)$ has petals that extend 5 units from the pole

Polar to Rectangular Conversion

These conversions rely on the relationships between the two coordinate systems:

$x = r\cos\theta, \quad y = r\sin\theta, \quad r^2 = x^2 + y^2, \quad \tan\theta = \frac{y}{x}$

Polar → Rectangular:

1. Multiply both sides of the polar equation by $r$ if needed (this lets you substitute $r^2 = x^2 + y^2$).
2. Replace $r\cos\theta$ with $x$ and $r\sin\theta$ with $y$.
3. Simplify.

For example, convert $r = 4\cos\theta$: multiply both sides by $r$ to get $r^2 = 4r\cos\theta$, then substitute to get $x^2 + y^2 = 4x$. Completing the square gives $(x-2)^2 + y^2 = 4$, which is a circle centered at $(2, 0)$ with radius 2.

Rectangular → Polar:

1. Replace $x$ with $r\cos\theta$, $y$ with $r\sin\theta$, and $x^2 + y^2$ with $r^2$.
2. Simplify and solve for $r$ in terms of $\theta$.

Converting between forms is useful because rectangular form makes it easy to spot standard shapes (circles, parabolas), while polar form reveals symmetry properties and simplifies curves like cardioids and roses that would be messy in rectangular coordinates.