Subsections

• Purpose
• Background
  • Cardioids, Limaçons, and Roses
  • Intersections of Curves in Polar Coordinates
  
  
• Maple Commands
• Area in Polar Coordinates
• Exercises

Graphs in Polar Coordinates

Purpose

The purpose of this lab is to help you become familiar with graphs in polar coordinates.

Background

There are many times in math, science, and engineering that coordinate systems other than the familiar one of Cartesian coordinates are convenient. In this lab, we consider one of the most common and useful such systems, that of polar coordinates.

The main reason for using polar coordinates is that they can be used to simply describe regions in the plane that would be very difficult to describe using Cartesian coordinates. For example, graphing the circle $x^2+y^2=1$ in Cartesian coordinates requires two functions - one for the upper half and one for the lower half. In polar coordinates, the same circle has the very simple representation $r=1$.

Cardioids, Limaçons, and Roses

These are three types of well-known graphs in polar coordinates. The table below will allow you to identify the graphs in the exercises.

Name	Equation
cardioid	$r = a(1 \pm \cos(\theta))$ or $r = a(1 \pm \sin(\theta))$
limaçon	$r = b \pm a \cos(\theta)$ or $r = b \pm a \sin(\theta)$
rose	$r=a \cos(n \theta)$ or $r=a \sin(n \theta)$

Intersections of Curves in Polar Coordinates

Finding where two graphs in Cartesian coordinates intersect is straightforward. You just set the two functions equal and solve for the values of $x$. In polar coordinates, the situation is more difficult. Most of the difficulties are due to the following considerations.

1. A point in the plane can have more than one representation in polar coordinates. For example, $r=1$, $\theta = \pi$ is the same point as $r=-1$, $\theta=0$. In general a point in the plane can have an infinite number of representations in polar coordinates, just by adding multiples of $2\pi$ to $\theta$. Even if you restrict $\theta$ a point in the plane can have several different representations.
2. The origin is determined by $r=0$. The angle $\theta$ can have any value.

These considerations can make finding the intersections of two graphs in polar coordinates a difficult task. As the exercises demonstrate, it usually requires a combination of plots and solving equations to find all of the intersections.

Maple Commands

>plot(cos(2*theta),theta=0..2*Pi,coords=polar);

Don't forget the option ,coords=polar! This graph is a four-leafed rose. Polar graphs can be hard to understand. Animating the graph as the angle increases will help.

>with(CalcP7):
>ParamPlot([cos(2*t),t],t=0..2*Pi,coords=polar);

When you run the ParamPlot command, you first get a set of axes with no curves drawn, and you think that there is something wrong. What you need to do to see the curves is first click on the graph. A box appears around the graph and a set of controls appears in the context bar just below the menu shortcut buttons at the top of the main Maple window. The set of controls works like those on a VCR. To see the animated graph, click on the play button. The other controls in the context bar allow you to slow down or speed up the animation, step through the animation one frame at a time, stop the animation, and even run the animation in reverse. I suggest you play with them until you feel comfortable. To find where two graphs intersect you set the functions equal to each other as they both equal the radius and then solve for the angle.

>plot([1+cos(theta),3/2],theta=0..2*Pi,coords=polar);
>plot([1+cos(theta),3/2],theta=0..2*Pi);

As discussed above there can be infinite solutions so use the fsolve command and choose a range of angles values in which the intersection point occurs.

>t1:=fsolve(1+cos(theta)=3/2,theta=0..2);
>t2:=fsolve(1+cos(theta)=3/2,theta=4..6);

The following commands find the radius vaue for each angle.

>1+cos(t1);
>1+cos(t2);

Area in Polar Coordinates

The relationship between area and integrals in polar coordinates is a little strange; the area inside a circle given (in polar coordinates) by $r = a$ is NOT just $$\int_{0}^{2\pi} r d\theta$$. Here is the rule: Area inside $r = f(\theta )$ is given by $$\frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 d\theta = \frac{1}{2} \int_{\theta_1}^{\theta_2} f(\theta )^2 d\theta$$. This comes from the fact that the area in a thin wedge with radius $r$ and angle $\Delta \Theta$ is $\frac{1}{2} r^2 \Delta \Theta$. Note that this gives you the right answer for a circle: $A = \pi r^2$. So to find the area of the cardiod use the following command.

>Area1:=1/2*int((1-cos(theta))^2, theta=0..2*Pi);
>evalf(Area1);

Exercises

1. For each of the following polar equations, plot the graph in polar coordinates using the plot command and identify the graph as a cardioid, limaçon, or rose.

   A)

   $r = \cos(4 \theta)$

   B)

   $r = 2 -3 \cos(\theta)$

   C)

   $r = 2 + 2\sin(\theta)$

2. Find all points of intersection for each pair of curves in polar coordinates.

   A)

   $r = 1+\cos(\theta)$ and $r=3/2$ for $0 \leq \theta \leq 2\pi$.

   B)

   $r = 2-\cos(\theta)$ and $r=3 \sin(\theta)$ for $0 \leq \theta \leq \pi$.

   C)

   $r=1+\sin(\theta)$ and $r=3 \cos(3 \theta)$ for $0 \leq \theta \leq \pi$.

3. A)

   Find the area of the region of the polar in exercise 2C that is inside the petal and outside the cardioid in the second quadrant.

   B)

   Plot the polar equation $r=3-5\sin(\theta)$. Find the angles that create the inner loop of $r=3-5\sin(\theta)$. Plot only the inner loop and then find the area inside the inner loop.