Subsections

• Background
• Getting Started
• Parametric Curves
• Plotting and animating curves in the plane
• Parametric curves
• Plotting Polar Curves
  • Cardioids, Limaçons, and Roses
  
  
• 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. You can access this worksheet by typing in the search field (magnifying glass next to start menu):

\\storage.wpi.edu\academics\math\calclab\MA1023\Parametric_polar_start_A19.mw

Parametric Curves

A parametric curve in the plane is represented by a pair of equations $x=f(t)$ and $y=g(t)$ for t in some interval I. A vector-valued function in the plane is a function $\mathbf{r}(t)$ that associates a vector in the plane with each value of $t$ in its domain. Such a vector valued function can always be written in component form as follows,

$$\mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j}$$

where $f$ and $g$ are functions defined on some interval $I$. From our definition of a parametric curve, it should be clear that you can always associate a parametric curve with a vector-valued function by just considering the curve traced out by the head of the vector.

Plotting and animating curves in the plane

The ParamPlot command is in the CalcP package so you have to load it first. If you get an error from this command, ask for help right away.

>with(CalcP7):

The ParamPlot command produces an animated plot. To see the animation, execute the command and then click on the plot region below to make the controls appear in the Context Bar just above the worksheet window. When the controls appear, you can click on the play button to animate the curve.

Parametric curves

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)

Both plots show that the given parametric curve is a circle. The plot command only shows the final curve whereas the ParamPlot command shows starting point, ending point and direction of motion.

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 $(r,\theta )$ where $x=r \cos(\theta)$ and $y=r \sin(\theta)$. 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

Other than circles, there 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 the circle $r=\cos(\theta )$ using a polar plot in Maple.

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

Exercises

1. Consider the simple function $y = x^{3} - x$. Animate the following two parameterizations $\bf {r}(t)=x(t) \bf {i} + y(t) \bf {j}$ for this curve for $-2 \leq t \leq 2$ and state how the two parameterizations of the same function are different.
   
   $$\mathbf{r}_1(t) = t \mathbf{i} + (t^{3} - t) \mathbf{j}$$
   
   
   
   
   $$\mathbf{r}_2(t) = -t \mathbf{i}-(t^{3} -t) \mathbf{j}$$
   
   

2. Find two different parametrizations of a semi-circle of radius 2, centered at the origin, one with and one without using trig functions for $x(t)$ and $y(t)$, the necessary interval of t for each, and plot each of the parametric curves on separate graphs using ParamPlot.

3. Plot each of the polar graphs below and identify the name of the polar graph using the table in the background.

   a)

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

   b)

   $$r = 1 - \sin (\theta )$$

   c)

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

4. 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? Find the angles that create only one petal of the rose $r=\sin(3\theta)$. Plot exactly one petal.

   b)

   Repeat part a with $r=\sin(2\theta)$.