Polar Coordinates and Polar Equations
This guide provides a thorough introduction to the polar coordinate system and polar equations. Learn key definitions, explore step-by-step worked examples, convert between Cartesian and polar forms, and visualize equations graphically.
Why use polar coordinates? While Cartesian coordinates are ideal for describing straight lines and rectangular shapes, polar coordinates excel at describing circular and rotational phenomena—such as planetary orbits, antenna radiation patterns, spiral galaxies, and the motion of a pendulum. Many curves that have complicated Cartesian equations (like the cardioid or rose curve) have simple, elegant polar equations.
Review of Cartesian (Rectangular) Coordinates
We begin by reviewing the Cartesian (rectangular) coordinate system. Let point \( A \) lie in the \( xy \)-plane. In Cartesian coordinates, point \( A \) is represented by an ordered pair \( (x, y) \), where:
- \( x \) is the signed distance from the \( y \)-axis
- \( y \) is the signed distance from the \( x \)-axis
In the figure below, the points \( (x,y) = (4,2) \) and \( (x,y) = (-3,4) \) are plotted.
Polar Coordinate System and Definitions
In the polar coordinate system, a point is represented by the ordered pair \( (r, \theta) \), where:
- \( r \) is the distance from the origin (called the pole)
- \( \theta \) is the angle measured from the polar axis
The value \( r \) is called the radial coordinate, and \( \theta \) is the angular coordinate.
For example, the points \( (5, \frac{\pi}{3}) \) and \( (4, \pi) \) are shown in the figure below.
By convention, the angle \( \theta \) is positive when measured counterclockwise and negative when measured clockwise.
Worked Examples
Example 1: Plotting Polar Points
Plot the points given by their polar coordinates:
a) \( (2,0) \) b) \( (2,\frac{3\pi}{4}) \) c) \( (4,\frac{7\pi}{3}) \) d) \( (3,-\frac{5\pi}{4}) \)
Solution to Example 1
The points are plotted in the polar grid below:
Note: Adding or subtracting integer multiples of \( 2\pi \) to the angular coordinate \( \theta \) while keeping \( r \) unchanged results in the same point:
\[ (r,\theta), \quad (r,\theta + 2\pi), \quad (r,\theta - 4\pi), \dots \]Example 2: Negative Radial Coordinates
The polar coordinate system allows the radial coordinate \( r \) to be negative. The points \( (-r, \theta) \) and \( (r, \theta) \) lie on the same line passing through the pole and are at the same distance \( |r| \) from the pole, but in opposite directions.
Therefore, the coordinates \( (-r, \theta) \) and \( (r, \theta + \pi) \) represent the exact same point.
Plot the following pairs of points:
a) \( (-2,0) \) and \( (2,0) \)
b) \( (-2,\frac{3\pi}{4}) \) and \( (2,\frac{3\pi}{4}) \)
c) \( (-4,-\frac{\pi}{3}) \) and \( (4,-\frac{\pi}{3}) \)
Solution to Example 2
Multiple Representations of a Point
Unlike Cartesian coordinates, polar coordinates are non-unique. A single point can be written in infinitely many ways. For any integers \( n \) and \( k \), the following represent the same point:
\[ (r,\theta), \quad (r,\theta + 2n\pi), \quad (-r,\theta + (2k+1)\pi) \]Relationships Between Polar and Cartesian Coordinates
Using basic trigonometry, we convert between polar \( (r, \theta) \) and rectangular \( (x, y) \) coordinates using:
\[ x = r\cos\theta, \qquad y = r\sin\theta \] \[ r^2 = x^2 + y^2, \qquad \tan\theta = \frac{y}{x} \quad (x \neq 0) \]
Polar Equations and Curves
A polar equation is expressed in the form \( f(r,\theta) = 0 \) or \( r = f(\theta) \). Its graph consists of all points \( (r,\theta) \) satisfying the equation.
Example 3: Graphing and Converting a Polar Equation
Sketch the graph of the polar equation \( r - 3\sin\theta = 0 \) and convert it to rectangular form.
Solution to Example 3
Step 1: Express \( r \) in terms of \( \theta \):
\[ r = 3\sin\theta \]Step 2: Construct a table of values:
| \( \theta \) | \( r = 3\sin\theta \) |
|---|---|
| \( 0 \) | \( 0 \) |
| \( \frac{\pi}{6} \) | \( 1.5 \) |
| \( \frac{\pi}{4} \) | \( \frac{3}{\sqrt{2}} \approx 2.12 \) |
| \( \frac{\pi}{3} \) | \( \frac{3\sqrt{3}}{2} \approx 2.60 \) |
| \( \frac{\pi}{2} \) | \( 3 \) |
| \( \frac{2\pi}{3} \) | \( \frac{3\sqrt{3}}{2} \approx 2.60 \) |
| \( \frac{3\pi}{4} \) | \( \frac{3}{\sqrt{2}} \approx 2.12 \) |
| \( \frac{5\pi}{6} \) | \( 1.5 \) |
| \( \pi \) | \( 0 \) |
Step 3: Convert to Cartesian coordinates:
Multiply both sides of \( r = 3\sin\theta \) by \( r \):
\[ r^2 = 3r\sin\theta \]Substitute \( r^2 = x^2 + y^2 \) and \( y = r\sin\theta \):
\[ x^2 + y^2 = 3y \implies x^2 + y^2 - 3y = 0 \]Complete the square for \( y \):
\[ x^2 + \left(y - \frac{3}{2}\right)^2 = \left(\frac{3}{2}\right)^2 \]This is the equation of a circle centered at \( (0, 1.5) \) with a radius of \( 1.5 \).
Common Polar Curves
Many classic curves have simple polar equations but complicated Cartesian equations. Here are the most important families:
Cardioid
Equation: \( r = a(1 \pm \cos\theta) \) or \( r = a(1 \pm \sin\theta) \)
The cardioid (from the Greek word for "heart") is a heart-shaped curve. It is commonly used in acoustics to model the directional response of certain microphones.
Limaçon
Equation: \( r = a \pm b\cos\theta \) or \( r = a \pm b\sin\theta \)
Depending on the ratio of \( a \) to \( b \), limaçons can be inner loops, dimpled, convex, or nearly circular. The cardioid is a special case where \( a = b \).
Rose Curve
Equation: \( r = a\cos(n\theta) \) or \( r = a\sin(n\theta) \)
Rose curves produce petal-like shapes. If \( n \) is odd, the curve has \( n \) petals; if \( n \) is even, it has \( 2n \) petals.
Lemniscate
Equation: \( r^2 = a^2\cos(2\theta) \) or \( r^2 = a^2\sin(2\theta) \)
The lemniscate is a figure-eight or infinity-shaped curve. It is a special case of the Cassini oval.
Frequently Asked Questions
How do you convert polar coordinates to Cartesian coordinates?
Use the formulas \( x = r\cos\theta \) and \( y = r\sin\theta \), where \( r \) is the radial distance and \( \theta \) is the angle from the polar axis.
How do you convert Cartesian coordinates to polar coordinates?
Use \( r^2 = x^2 + y^2 \) to find the radial distance, and \( \tan\theta = \frac{y}{x} \) to find the angle. Be sure to choose the correct quadrant for \( \theta \) based on the signs of \( x \) and \( y \).
Can the radial coordinate r be negative in polar coordinates?
Yes. A negative \( r \) means the point lies in the opposite direction of the angle \( \theta \). The point \( (-r, \theta) \) is the same as \( (r, \theta + \pi) \).
Are polar coordinates unique?
No. Unlike Cartesian coordinates, a single point can be represented by infinitely many polar coordinate pairs. For example, \( (r, \theta) \), \( (r, \theta + 2\pi) \), and \( (-r, \theta + \pi) \) all represent the same point.
What is a cardioid in polar coordinates?
A cardioid is a heart-shaped curve given by equations of the form \( r = a(1 + \cos\theta) \) or \( r = a(1 - \cos\theta) \), where \( a \) is a positive constant.
Calculators & Further Reading
Polar Coordinates Tutorial | Polar to Rectangular Calculator | Rectangular to Polar Calculator