Polar Curves
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Section 7.3: “Polar Coordinates” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/7-3-polar-coordinates |
| Textbook used in class | Stewart, Calculus, Section 10.3: “Polar Coordinates” |
Opening Scenario
The equation $r = 1 + \cos\theta$ describes a heart-shaped curve called a cardioid. The equation $r = \sin(2\theta)$ traces a four-petaled rose. Neither of these has a simple Cartesian form, but they are natural and elegant in polar coordinates. Plotting polar curves requires building a table of $(r, \theta)$ values, then converting to Cartesian points or plotting on a polar grid.
Quick Reference
Graphing $r = f(\theta)$: Choose key values of $\theta$, compute $r$, plot the points $(r,\theta)$ on a polar grid. Connect in order of increasing $\theta$.
Common polar curves:
| Curve | Equation | Description |
|---|---|---|
| Circle (centered at origin) | $r = a$ | Radius $a$ |
| Circle (through origin) | $r = 2a\cos\theta$ | Center $(a,0)$, radius $a$ |
| Cardioid | $r = a(1+\cos\theta)$ | Heart shape; passes through origin |
| Limaon | $r = a + b\cos\theta$ | Varies by ratio $a/b$ |
| Rose curve | $r = a\cos(n\theta)$ | $n$ petals if $n$ odd; $2n$ petals if $n$ even |
| Lemniscate | $r^2 = a^2\cos(2\theta)$ | Figure-eight curve |
| Archimedean spiral | $r = a\theta$ | Spiral expanding from origin |
Key Concepts
1. Building a Table for $r = f(\theta)$
Choose $\theta = 0, \pi/6, \pi/4, \pi/3, \pi/2, \pi, 3\pi/2, 2\pi$ (and values where $r = 0$ or $r$ is maximum). Compute $r = f(\theta)$ at each. Plot $(r,\theta)$ on a polar grid or convert to Cartesian $(r\cos\theta, r\sin\theta)$.
When $r < 0$ at some $\theta$, the point is in the direction $\theta + \pi$. These points are still part of the curve.
2. Symmetry Tests
Polar symmetry can be detected before plotting:
| Symmetry | Test | Meaning |
|---|---|---|
| About $x$-axis | $f(-\theta) = f(\theta)$ (or $f(\pi-\theta) = f(\theta)$) | Reflects across $x$-axis |
| About $y$-axis | $f(\pi-\theta) = f(\theta)$ | Reflects across $y$-axis |
| About origin | $f(\theta+\pi) = f(\theta)$ or $f(-\theta) = -f(\theta)$ | 180-degree rotational symmetry |
Warning: The symmetry tests are sufficient but not necessary -- a polar curve can have symmetry even if none of the algebraic tests hold (due to multiple representations of the same point in polar).
3. Key Features of Common Curves
Cardioid $r = a(1+\cos\theta)$:
- Maximum $r = 2a$ at $\theta = 0$.
- Zero at $\theta = \pi$ (passes through origin).
- Symmetric about $x$-axis.
Rose curve $r = a\cos(n\theta)$:
- $n$ petals if $n$ is odd ($n = 1, 3, 5, \ldots$): each petal traced as $\theta$ sweeps a half-range.
- $2n$ petals if $n$ is even.
- Tips at $r = a$, origin when $\cos(n\theta) = 0$.
Worked Example
Sketch the cardioid $r = 1 + \sin\theta$.
Table of values:
| $\theta$ | $r = 1+\sin\theta$ |
|---|---|
| $0$ | 1 |
| $\pi/6$ | 1.5 |
| $\pi/2$ | 2 |
| $5\pi/6$ | 1.5 |
| $\pi$ | 1 |
| $3\pi/2$ | 0 |
| $2\pi$ | 1 |
Key features:
- Maximum $r = 2$ at $\theta = \pi/2$ (top).
- $r = 0$ at $\theta = 3\pi/2$ (curve passes through origin at the bottom).
- Symmetric about the $y$-axis (since $\sin(\pi-\theta) = \sin\theta$, so $f(\pi-\theta) = 1+\sin(\pi-\theta) = 1+\sin\theta = f(\theta)$).
Description: A heart shape pointing upward (cardioid). It starts at $(1,0)$ on the right, expands to the top at $(0,2)$ in Cartesian coordinates, contracts to the origin at the bottom, and returns to the right. The curve is traced once as $\theta$ goes from $0$ to $2\pi$.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Ignoring $r < 0$ points | Stopping the curve when $r$ first becomes negative | Points with $r < 0$ lie in the direction $\theta + \pi$; they are part of the curve |
| Plotting $r$ vs. $\theta$ as a Cartesian graph | Drawing a sine wave instead of a cardioid | $r = 1+\sin\theta$ is plotted on a polar grid, not on $(r,\theta)$ axes |
| Missing petals of a rose curve | Drawing 2 petals of $r = \cos(2\theta)$ instead of 4 | For even $n$, $r = a\cos(n\theta)$ has $2n$ petals; trace over $[0,2\pi]$ to capture all of them |
Common Misconceptions
the graph of $r = f(\theta)$ is drawn on $(r, \theta)$ axes like a Cartesian graph.
This is the iconic-graph error. The polar equation $r = 1 + \cos\theta$ describes a curve in the plane with the origin at the pole. Plotting $r$ on the vertical axis against $\theta$ on the horizontal axis produces a sine-wave-like picture that is not the cardioid. The correct picture plots each point $(r\cos\theta, r\sin\theta)$ in the plane for the appropriate $\theta$-values.
when $r < 0$ for some $\theta$, those points are not part of the curve.
This is the concept-image-conflicts-definition error. A polar curve is defined by $r = f(\theta)$ for all $\theta$ in the given range, including values where $f(\theta) < 0$. A negative $r$ places the point in the direction $\theta + \pi$. Ignoring those points omits entire petals from rose curves and entire branches from limacons.
Leveled Practice
Level 1 -- Build a Table and Identify
Problem 1. Build a table of values and identify the curve $r = 3\cos\theta$, $0 \leq \theta \leq \pi$.
Show answer
| $\theta$ | $r = 3\cos\theta$ | $(x,y)$ |
|---|---|---|
| 0 | 3 | $(3,0)$ |
| $\pi/3$ | 1.5 | $(0.75, 1.3)$ |
| $\pi/2$ | 0 | $(0,0)$ |
| $2\pi/3$ | $-1.5$ | $(0.75,-1.3)$ |
| $\pi$ | $-3$ | $(3,0)$ |
Convert: $x = r\cos\theta = 3\cos^2\theta$, $y = r\sin\theta = 3\cos\theta\sin\theta$.
Cartesian: $x^2+y^2 = r^2 = 3r\cos\theta = 3x$, so $(x-3/2)^2+y^2 = 9/4$. This is a circle of radius $3/2$ centered at $(3/2, 0)$.
Level 2 -- Count Petals
Problem 2. How many petals does $r = \cos(3\theta)$ have, and for which $\theta$-values do the petals occur?
Show answer
$n = 3$ is odd, so $r = \cos(3\theta)$ has $n = 3$ petals.
Petals occur where $r$ is maximum ($r = 1$), i.e., $3\theta = 0, 2\pi, 4\pi$, giving $\theta = 0, 2\pi/3, 4\pi/3$.
The petal tips are at $\theta = 0, 2\pi/3, 4\pi/3$.
Mastery Checklist
Mental Model
A polar curve $r = f(\theta)$ is traced by a radius that rotates counterclockwise from $\theta = 0$. At each angle, the length of the radius is $r = f(\theta)$. As $\theta$ sweeps around, the tip of this rotating radius traces the curve. Where $f(\theta) = 0$, the tip is at the origin. Where $f(\theta)$ is maximum, the tip is farthest from the origin. Symmetry means the curve “mirrors” itself across some axis.
Connections
Looking back
- Polar coordinates (Section 10.3): The $(r,\theta)$ system is the foundation.
Looking ahead
- Polar area (Section 10.4): Area swept by the rotating radius as $\theta$ changes.
- Polar tangent lines (Section 10.4): Tangent direction computed from parametric differentiation.