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

cp /math/calclab/MA1023/Parametric_start_C08.mws ~/My_Documents

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_C08.mws`, and go through it
carefully. Then you can start working on the exercises.

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.

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 before for the lab
on polar coordinates. Examples are in the `Getting Started`
worksheet.

It is clear that this formula doesn't make sense if at some particular value of . If at that same value of , then it turns out the graph has a vertical tangent at that point. If both and are zero at some value of , then the curve often doesn't have a tangent line at that point. What you see instead is a sharp corner, called a cusp.An example of this appears in the first exercise.

While the concept of arc length is very useful for the theory of parametric curves, it turns out to be very difficult to compute in all but the simplest cases.

>with(plots): >with(CalcP7): >implicitplot(x^2=y,x=-2..2,y=0..4,scaling=constrained); >ParamPlot([t,t^2],t=-2..2,scaling=constrained); >ParamPlot([-t,t^2],t=-2..2,scaling=constrained);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. To enter a function parametrically

>f:=t->[t*cos(3*t),t^2]; >VPlot(f(t),t=-2*Pi..2*Pi);

- 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 . The sharp points in the graph are called cusps. Use the formula for the slope of a parametric curve to find values of for where these cusps occur and explain why it makes sense for the cusps to occur only at these values of . That is, verify that the curve has a slope at all other values of in the interval. - The family of parametric curves

where and are positive integers, is an example of what is called a Lissajous figure. Use a parametric plot to plot the three cases , and and describe what you see. - The parametric description , ,
is the ellipse

First show that the two are the same shape by plotting them parametrically and with the command`implicitplot`. Use the formula above to set up an integral for the arc length of the ellipse. You should find that Maple can't do the integral exactly. This is because this integral can't be done analytically. You can get a numerical approximation to the integral by putting an`evalf`command on the outside of the`int`command. - Consider the family of parametric curves
and
, where and are positive constants, with and is a positive integer. Use a paramentric plot to test the following properties of this family.
- i
- The curve only has loops in it if . (Plot the example in the worksheet to see what loops look like.)
- ii
- If the curve has loops in it, the number of loops is .
- iii
- The curve has cusps if and there are exactly of them. Use at least two sets of values of and to test each property. Don't forget that has to be an integer and should be smaller than 20.

2008-02-12