Polar graphing

Understanding Polar Graphs

In the parent chapter on polar coordinates, you see points described by a radius $r$ and an angle $\theta$. Polar graphing is about how to draw and interpret curves given by relationships between $r$ and $\theta$.

In polar graphing:

• The horizontal positive $x$-axis is the polar axis (angle $\theta = 0$).
• Angles increase counterclockwise.
• A point is plotted by:
1. Rotating from the polar axis by angle $\theta$.
2. Moving $|r|$ units from the origin along that direction.
3. If $r < 0$, you move in the opposite direction of the angle.

In this chapter we focus on equations involving $r$ and $\theta$, and what their graphs look like.

Basic Types of Polar Equations

Polar equations usually relate $r$ and $\theta$ in one of these ways:

• $r = f(\theta)$
• $\theta = \text{constant}$
• Equations involving both $r$ and $\theta$ that can sometimes be converted to Cartesian form, or vice versa.

Here are key families of curves that are characteristic of polar graphing.

1. Circles in Polar Form

Some circles have especially simple polar equations.

Circles centered at the origin

A circle of radius $a$ centered at the origin has the polar equation
$$
r = a
$$
with $a > 0$.

To sketch $r = a$:

• For any angle $\theta$, the radius is always $a$.
• So every point is $a$ units from the origin.
• The result is a circle centered at the origin with radius $a$.

Circles not centered at the origin (simple case)

A common polar form of a circle is
$$
r = 2a\cos\theta \quad\text{or}\quad r = 2a\sin\theta,
$$
with $a > 0$. These describe circles whose centers lie on the axes.

• $r = 2a\cos\theta$ is a circle of radius $a$ whose center is at $(a,0)$ in Cartesian coordinates.
• $r = 2a\sin\theta$ is a circle of radius $a$ whose center is at $(0,a)$.

You do not need to derive these here; what matters is to recognize them as circles off the origin and know their rough shapes.

Lines in Polar Coordinates

Some very simple lines in Cartesian coordinates appear naturally in polar form:

• $\theta = \alpha$ (where $\alpha$ is constant) represents a ray from the origin at angle $\alpha$.
• It is like a straight line through the origin, but often we only plot the part with $r \ge 0$.
• $r\cos\theta = a$ or $r\sin\theta = b$ may describe vertical or horizontal lines, since
  $x = r\cos\theta$ and $y = r\sin\theta$.

For graphing practice in polar coordinates, it is valuable to:

• Recognize that $\theta = \text{constant}$ gives a straight line through the origin.
• Understand that $r = \text{constant}$ gives a circle centered at the origin.

Graphing $r = f(\theta)$ by Plotting Points

The basic method of polar graphing uses a table of values.

Suppose you want to graph $r = f(\theta)$:

1. Choose a range of $\theta$ values.
• Many interesting polar curves are periodic in $\theta$.
• Common intervals:
• $[0, 2\pi]$ (one full rotation),
• sometimes $[0, \pi]$ or even smaller intervals are enough.
2. Compute corresponding $r$ values.
• For each chosen $\theta$, calculate $r = f(\theta)$.
• Include positive, zero, and possible negative $r$ values.
3. Plot each point.
• For each pair $(r, \theta)$, plot using the polar coordinate rules:
• Rotate by $\theta$ from the polar axis.
• Move $|r|$ units from the origin.
• If $r < 0$, go in the direction opposite angle $\theta$.
4. Connect the points smoothly.
• Use the behavior of $r$ as $\theta$ changes to see the direction in which the curve is traced.
• Consider symmetry (see below) to fill in the rest of the graph efficiently.

This method is slower than recognizing special forms, but it works for any polar equation.

Symmetry in Polar Graphs

Symmetry helps you graph more quickly and with fewer calculations. For a polar equation, you can test for symmetries by substituting related angles and radii.

Let the polar equation be written as a relation between $r$ and $\theta$. Typical symmetries:

1. Symmetry about the polar axis (the $x$-axis)
• Replace $\theta$ by $-\theta$.
• If the equation is unchanged, the graph is symmetric about the polar axis.
2. Symmetry about the line $\theta = \frac{\pi}{2}$ (the $y$-axis)
• Replace $\theta$ by $\pi - \theta$.
• If the equation is unchanged, the graph is symmetric about $\theta = \frac{\pi}{2}$.
3. Symmetry about the origin
• Replace $r$ by $-r$ and $\theta$ by $\theta + \pi$ (this represents the same Cartesian point).
• Equivalently, check whether replacing $\theta$ with $\theta + \pi$ leaves the equation unchanged.
• If the equation is unchanged under this change, the graph is symmetric about the origin.

These tests are often used with standard forms like $r = a\sin\theta$ and $r = a\cos\theta$ to quickly see their shape.

Special Polar Curves

Many of the most important and picturesque polar graphs belong to a few families with standard forms. Here you learn what those forms look like; more detailed study of each type can come later.

1. Cardioids and Limacons

These curves generally have the form
$$
r = a \pm b\cos\theta \quad\text{or}\quad r = a \pm b\sin\theta
$$
with $a > 0$, $b > 0$.

They are related, but their shapes depend on the ratio $\frac{a}{b}$.

Cardioid

A cardioid (heart-shaped curve) occurs when
$$
a = b.
$$
For example:

• $r = a(1 + \cos\theta)$,
• $r = a(1 + \sin\theta)$.

Characteristics:

• Has a single cusp (a sharp point) at the origin.
• Symmetric about the polar axis for $\cos$-forms.
• Symmetric about the line $\theta = \frac{\pi}{2}$ for $\sin$-forms.

Limacons

When $a$ and $b$ are not equal, you get a limacon, which can have:

• An inner loop ($a < b$),
• A dimpled shape ($a$ slightly larger than $b$),
• A nearly circular shape ($a \gg b$).

Different cases:

• Inner loop: $0 < a < b$.
• Cardioid: $a = b$.
• Dimpled limacon: $b < a < 2b$.
• Convex limacon: $a \ge 2b$.

Graphing strategy:

• Use symmetry.
• Check values at special angles like $\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$.
• Look for angles where $r = 0$; these often correspond to special features (like loops).

2. Rose Curves

Rose curves look like flowers with several petals. Their standard forms are
$$
r = a\cos(n\theta) \quad\text{or}\quad r = a\sin(n\theta),
$$
where $a > 0$ and $n$ is a positive integer.

Number of petals:

• If $n$ is odd, the graph has $n$ petals.
• If $n$ is even, the graph has $2n$ petals.

Examples:

• $r = \cos(3\theta)$ → $3$-petal rose.
• $r = 2\sin(4\theta)$ → $8$-petal rose.

Graphing hints:

• The maximum radius is $|a|$ (when $\cos(n\theta)$ or $\sin(n\theta)$ equals $\pm 1$).
• The curve repeats as $\theta$ changes by $\frac{\pi}{n}$ or $\frac{2\pi}{n}$.
• Often, graphing on $[0, \pi]$ or even $[0, \frac{\pi}{n}]$ is enough to see one full pattern because of periodicity.

3. Lemniscates

A lemniscate has a figure-eight shape (like an infinity symbol). Standard forms include
$$
r^2 = a^2\cos(2\theta) \quad\text{or}\quad r^2 = a^2\sin(2\theta),
$$
with $a > 0$.

Key features:

• The equation involves $r^2$, so for each $\theta$ you have
  $$
  r = \pm a\sqrt{\cos(2\theta)} \quad\text{or}\quad r = \pm a\sqrt{\sin(2\theta)},
  $$
  whenever the quantity inside the square root is nonnegative.
• The curve often has two symmetric loops.
• Symmetry:
• $\cos(2\theta)$ form is symmetric about the polar axis and the origin.
• $\sin(2\theta)$ form is symmetric about the lines $\theta = \frac{\pi}{4}$ and $\theta = \frac{5\pi}{4}$.

Graphing:

• Determine the angles where $\cos(2\theta)$ or $\sin(2\theta)$ is positive or zero.
• Within those intervals, compute a few radii and plot both $+r$ and $-r$ where appropriate to see both loops.

4. Spirals

While many spirals can be expressed in polar form, a classic example is the Archimedean spiral:
$$
r = a\theta,
$$
with $a > 0$.

As $\theta$ increases:

• The radius $r$ increases proportionally.
• The curve winds outward from the origin, crossing each radial line a fixed distance farther out each time.

Graphing:

• Choose several values of $\theta$ (e.g. $0, \frac{\pi}{4}, \frac{\pi}{2}, \pi, 2\pi,\dots$).
• Calculate corresponding $r$ and plot.
• Notice how the curve keeps moving farther away as $\theta$ grows.

Relationship Between Tracing and $\theta$

When graphing $r = f(\theta)$, the direction in which the curve is traced as $\theta$ increases is often important:

• As you increase $\theta$ from a starting value, plot points in order; this shows how the curve is drawn.
• For some curves (like roses and lemniscates), the same physical points can be visited more than once as $\theta$ continues beyond a basic interval.
• Often, you only need to graph over one fundamental interval of $\theta$ where the pattern doesn’t repeat. For example:
• For many rose curves with $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$, the full graph can appear over $[0, \pi]$ or less.

Knowing the period of $f(\theta)$ helps:

• If $f$ has period $T$ (so $f(\theta + T) = f(\theta)$), then the polar curve usually repeats its pattern every $T$ radians in $\theta$, possibly with some overlap due to the $r < 0$ interpretation.

Choosing a Good $\theta$ Interval

When you are asked to sketch or analyze a polar graph, think about:

1. Period of the function in $\theta$.
• For example, $\sin(n\theta)$ and $\cos(n\theta)$ have period $\frac{2\pi}{n}$.
2. Simplest interval covering at least one full pattern of the curve.
3. Symmetry, to extend a partial graph to the whole picture.

Common choices:

• Rose curves: sometimes $[0, \pi]$ or $[0, \frac{\pi}{n}]$.
• Cardioids and limacons: usually $[0, 2\pi]$.
• Lemniscates: sometimes $[-\frac{\pi}{4}, \frac{\pi}{4}]$ or similar, depending on the form.

Converting Between Polar and Cartesian for Graphing

Sometimes you are given an equation in Cartesian form but want to recognize the polar curve. The conversions (introduced in the parent chapter) are:

• $x = r\cos\theta$,
• $y = r\sin\theta$,
• $r^2 = x^2 + y^2$,
• $\tan\theta = \dfrac{y}{x}$ (with attention to quadrants).

In polar graphing, you may:

• Convert a Cartesian equation to polar to see if it matches one of the familiar forms (circle, line, etc.).
• Convert a polar equation to Cartesian to recognize the underlying curve in a more familiar coordinate system, even if you still choose to graph it using polar methods.

For example, starting from $r = 2a\cos\theta$:

• Multiply both sides by $r$: $r^2 = 2ar\cos\theta$.
• Replace $r^2$ by $x^2 + y^2$ and $r\cos\theta$ by $x$:
  $$
  x^2 + y^2 = 2ax.
  $$
• This reveals a circle in Cartesian form and explains its position.

You do not need to perform such conversions every time you graph, but recognizing them helps you understand what you are graphing.

Practical Strategy for Sketching a Polar Graph

When you see a polar equation and are asked to sketch its graph, a reasonable strategy:

1. Identify the type (if possible):
• Is it a circle, line, cardioid, limacon, rose, lemniscate, spiral, or something else?
2. Check for symmetries using substitution:
• About the polar axis, about $\theta = \frac{\pi}{2}$, or about the origin.
3. Find key points:
• Where $r = 0$.
• Where $r$ is maximal or minimal.
• Values at simple angles like $0, \frac{\pi}{4}, \frac{\pi}{2}, \pi$, etc.
4. Decide the $\theta$ interval you need:
• Often one or two periods of $f(\theta)$ are enough.
5. Plot a table of $(\theta, r)$ values over that interval:
• Include enough points to see the curve’s shape, especially near interesting features (cusps, loops).
6. Draw the curve smoothly, using symmetry and periodicity to complete the picture beyond your plotted points.

By combining recognition of standard forms, symmetry tests, and point-plotting, you can graph a wide variety of polar equations accurately and efficiently.