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The purpose of this lab is to give you practice with parametric
curves in the plane and in visualizing parametric curves as
To assist you, there is a worksheet associated with this lab that
contains examples and even solutions to some of the exercises. You can
copy that worksheet to your home directory with the following command,
which must be run in a terminal window, not in Maple.
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.
A parametric curve in the plane is 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
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
The graph of a parametric curve may not have a slope at every point on
the curve. When the slope exists, it must be given by the formula
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
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.
As mentioned above, parametric curves often represent the motion of a
particle or mechanical system. As we will see in class, when we think
of a parametric curve as representing motion, we need a way to measure
the distance traveled by the particle. This distance is given by the
arc length, , of a curve. For a parametric curve ,
, the arc length of the curve for
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.
There are a variety of ways to work with parametric equations in Maple. There is an animation command that shows how the graph is plotted over t. For example the parabola can be written parametrically in different ways two of them are and
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
- The cycloid is a famous example of a parametric curve having
several important applications. Use the ParamPlot command to
animate the cycloid
. 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
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
, where and are positive constants, with and is a positive integer. Use a paramentric plot to test the following properties of this family.
- The curve only has loops in it if . (Plot the example in the worksheet to see what loops look like.)
- If the curve has loops in it, the number of loops is .
- 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.
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