6.4. Graphs of Polar Equations
Note: All images in this section were taken from Wikipedia
In the previous section, we discussed the polar coordinate system at length as well as how to convert back and forth between rectangular and polar coordinates. In this section, we will discuss how to graph some simple relations (functions of the polar angle θ) in polar coordinates and what these graphs look like.
All graphs we consider in this chapter have the form r = f(θ), that is, the radial coordinate is a function of the angular coordinate, just as the functional graphs we looked at in rectangular coordinates have the form y = f(x). Thus we see that the radial coordinate, r, is now taking the place of the rectangular coordinate y and that the angular coordinate θ is replacing the role of x. As we will see, this replacement will lead to some interesting graphs.
The most basic graph in polar coordinates is a circle centered around the origin. The equation of such a graph is simply r = R, where R is a positive constant (the radius of the circle). By virtue of the Equation (6.3.3a), we see that the equivalent relation in rectangular coordinates is √x2 + y2 = R, or x2 + y2 = R2, agreeing with Equation (1.4.1). The rectangular coordinate analog of the circle in polar coordinates is a horizontal line (constant linear function).
Besides circles centered about the origin, circles containing the origin have simple equations in polar coordinates. Consider the polar equation r = a cos θ, where a is a positive constant. Multiplying both sides by r, we obtain r2 = ar cos θ, which in rectangular coordinates becomes x2 + y2 = ax. Note that we can rewrite this equation as (x2 - ax) + y2 = 0. Next we complete the square on the terms in parentheses by adding a2/4 to both sides, obtaining (x - a/2)2 + y2 = (a/2)2. Now we recognize this as the equation of a circle centered at the point (a/2, 0) with radius a/2. Similarly, the equation r = a sin θ represents a circle centered at the point (0, a/2) with radius a/2. Note that both of these circles are traversed twice as θ varies from 0 to 2π.
Lines also have simple equations in polar coordinates. Consider the form Ax + By = C for the general equation of a line in the xy-plane. From the coordinate transformation (6.3.2), we see that this equations becomes
- (6.4.1) r(A cos θ + B sin θ) = C
Note that horizontal lines have the form y = C, which in polar coordinates becomes r sin θ = C, and that vertical lines have the form x = C, which in polar coordinated becomes r cos θ = C.
The next most basic graph in polar coordinates, the analog of a line in rectangular coordinates, is the Archimedean spiral. This graph has the form
- (6.4.2) r = a + bθ
where a and b are arbitrary real-valued constants. As it turns out, all Archimedean spirals are similar, i.e. they have the same shape. Below we show a graph of the Archimedean spiral given by the equation r = θ.
Figure 6.4.3: Archimedean Spiral r = θ
Perhaps even more interesting than the Archimedean spiral are logarithmic spirals. These remarkable curves have the formula
- (6.4.4) r = a * ebθ
where a and b are arbitrary nonzero constants with a positive and e ≈ 2.71828 is the base of natural logarithms, introduced briefly in Chapter 3. A cross-section of the shell of a chambered nautilus reveals a logarithmic spiral. The well-renowned German mathematician Jakob Bernoulli studied these spirals in detail, discovering many fascinating properties of these curves.
Figure 6.4.5: Logarithmic Spiral
Another very interesting figure which has a simple equation in polar coordinates but not in rectangular coordinates is the cardioid. There are four basic forms of the equation of a cardioid in polar coordinates. They are as follows:
- (6.4.6a) r = a(1 + cos θ)
- (6.4.6b) r = a(1 + sin θ)
- (6.4.6c) r = a(1 - cos θ)
- (6.4.6d) r = a(1 - sin θ)
Figure 6.4.7: The Four Basic Cardioids
Each of these forms has a different orientation as shown. Note that a cardioid has a heartlike shape, hence the name. Readers familiar with the Mandelbrot set may notice that the main lobe of this set is a cardioid.
Perhaps some of the prettiest graphs in polar coordinates are polar roses, also known as rhodonea curves. The equations of these fascinating figures have the simple form
- (6.4.8) r = cos kθ
where k is an arbitrary rational number. (If k is irrational, the curve never closes.) It turns out that if k is an integer, then the resulting polar rose looks like a flower with k petals if k is odd or 2k petals if k is even. If k is not an integer, then the curve intersects itself. The following figure shows all with k = n/d for n and d both ranging from 1 to 7.
Figure 6.4.9: Polar Roses