The purpose of this lab is to give you practice with parametrizing curves in the plane and in visualizing parametric curves as representing motion.

cp ~bfarr/Parametric_start.mws ~

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

A parametric curve in the plane can be defined as an ordered pair, , of functions, with representing the coordinate and the coordinate. Parametric curves arise naturally as the solutions of differential equations and often represent the motion of a particle or a mechanical system. They also often arise in studying oscillations in electrical circuits.

For example, neglecting air resistance, the position of a projectile fired from the origin at an initial speed of and angle of inclination is given by the parametric equations

where is time and is the acceleration due to gravity.

Graphically, a parametric curve can be represented several ways. One simple way is to plot the component functions, and , individually versus the independent variable . Another way is to plot the set of points . This gives you the curve along which the particle moves, but information on how it moves has been lost. On the other hand, plotting the component functions individually makes it hard to see how the particle is actually moving.

To help you to visualize parametric curves as representing motion, a
Maple routine called `ParamPlot` has been written. It uses the
Maple `animate` command to actually show the particle moving along
its trajectory. You actually used this command last term for the lab
on polar coordinates. Examples are in the `Getting Started`
worksheet.

By restricting to an interval , you can get a parametric description of a portion of the curve. For example, the right half of the parabola would result from for .

More complicated parametrizations of can be obtained with
parametric curves of the form
. By choosing
appropriately, for example, you can make the particle stop and turn back on the
curve. For example, suppose that the curve to be parametrized is the
graph of the function . The following examples give three
different parametrizations of parts of this curve.

- The cycloid is a famous example of a parametric curve having
several important applications. Use the
`ParamPlot`command to animate the cycloid , over the interval and then use the`plot`command to generate a printable plot of this cycloid over the same interval. - The family of parametric curves

where and are positive integers, is an example of what is called a Lissajous figure. Use`ParamPlot`to plot the two cases and and describe what you see. Include printable plots of each curve in your printout. - The last example, concerning the straight line
, provided three different parametrizations. Match the
parametrizations with the descriptions below. Justify your choices.
- Starts at the point . Moves to the left initially, but reverses direction once.
- Starts at the origin, then oscillates between the points and .
- Starts at the point and then moves to the right along the line.

- Given the function , find a function that
makes the parametric description
do the
following. Make sure that you explain your choices.
- Goes through the origin at and traverses the curve from right to left as increases.
- Goes through the origin at and oscillates between the points and .

2001-10-29