Subsections

• Background
• Getting Started
• Parametric curves and their Slope
• The parametric form of a line
• Plotting Polar Curves
  • Cardioids, Limaçons, and Roses
  
  
• Area in Polar Coordinates
• Exercises

Parametric Curves and Polar Coordinates

Background

The use of parametric equations and polar coordinates allows for the analysis of families of curves difficult to handle through rectangular coordinates. If a curve is a rectangular coordinate graph of a function, it cannot have any loops since, for a given $x$ value there can be at most one corresponding $y$ value. However, using polar coordinates, curves with loops can appear as graphs of functions.

Getting Started

To assist you, there is a worksheet associated with this lab that contains examples similar to some of the exercises.On your Maple screen go to File - Open then type the following in the white rectangle:

\\storage\academics\math\calclab\MA1023\Parametric_polar_start_C19.mw

You can copy the worksheet now, but you should read through the lab before you load it into Maple. Once you have read through the exercises, start up Maple, load the worksheet, and go through it carefully. Then you can start working on the exercises.

Parametric curves and their Slope

For a curve defined parametrically by the equations $x=f(t)$ and $y=g(t)$,

> f:=t->cos(t)
> g:=t->sin(t)

The parametric curve can be plotted with or without animation:

> with(CalcP7):
> plot([f(t),g(t),t=0..2Pi])
> ParamPlot([f(t),g(t)],t=0..2Pi)

The graph of a parametric curve may not have a slope at every point on the curve. When the slope exists, it is given by the formula:

$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

In Maple, the slope of a parametric curve can be calculated using the $\texttt{diff}$ command as in the example below.

> m:=eval((diff(g(t),t))/(diff(f(t),t)),t=Pi/4)

The parametric form of a line

The parametric equations for a line passing through a point $(x1,y1)$ having slope $m$ is given by:

$$x(t)=x1+t, \hspace{5ex} y(t)=y1+mt, \hspace{5ex} -\infty < t < \infty$$

You can evaluate $x$ and $y$ coordinates at any value of $t$ in the usual way and define the parametric equations for the tangent line at the given $t$ values as follows:

> x1:=f(Pi/4)
> y1:=g(Pi/4)
> fline:=t->x1+t
> gline:=t->y1+mt
> plot([[f(t),g(t),t=0..2Pi],[fline(t),gline(t),t=-1..1]])

Plotting Polar Curves

When you graph curves in polar coordinates, you are really working with parametric curves. The basic idea is that you want to plot a set of points by giving their coordinates in $(x,y)$ pairs. When you use polar coordinates, you are defining the points $(x,y)$ in terms of polar coordinates $(r,\theta )$. When you plot polar curves, you are usually assuming that $r$ is a function of the angle $\theta$ and $\theta$ is the parameter that describes the curve. In Maple you have to put square brackets around the curve and add the specification coords=polar. Maple assumes that the first coordinate in the parametric plot is the radius $(r)$ and the second coordinates is the angle $(\theta )$.

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 $r=a \sin(n \theta)$

Below is an example of how to plot and animate a cardioid using a polar plot in Maple.

> plot(1-cos(theta),theta=0..2Pi,coords=polar)
> plots[animate](plot,[cos(theta),theta=0..t,coords=polar],t=0..2Pi)

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.

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

Exercises

1. Consider the parametric curve $x(t)=6\cos(t)+\cos(6t)$ and $y(t)=6\sin(t)+\sin(6t)$ for $0 \leq t \leq 2\pi$.

   a)

   Plot the parametric curve with and without animation.

   b)

   Describe what effect negating the angle has on animation. Describe what effect doubling the anle has on animation.

   c)

   Calculate the slope and the $(x,y)$ ordered pair at $$t = \frac{7 \pi}{6}$$.

   d)

   Define the parametric equation of the line tangent to the curve at that point and plot the parametric curve and this tangent line on the same graph over the interval $0 \leq t \leq 2\pi$ for the curve and $-3 \leq t \leq 3$ for the line. You do NOT need to animate this plot.

2. Plot the given pair of curves on the same graph in polar coordinates and in rectangular coordinates without animation. Find all polar points of intersection.
   
   
   $$r1 = 3-2\cos(\theta), \hspace{5ex} r2=4\sin(3\theta), \hspace{5ex} 0 \leq \theta \leq 2\pi$$
   
   

3. a)

   Animate the plot of the three-petal rose $r=\sin(3\theta)$ in polar coordinates over the interval $0 \leq \theta \leq \pi$ and again over the interval $0 \leq \theta \leq 2\pi$. What is the necessary interval of $\theta$ values needed to traverse the polar plot exactly once?

   b)

   Find the angles that create only one petal of the rose $r=\sin(3\theta)$. Plot only one petal and find the area of that petal. Find the area of the entire plot with and without symmetry.