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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.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.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.
- 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 at
, , and are called cusps. Use the formula for the
slope of a parametric curve to 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 ParamPlot to plot the two cases
and and describe what you see.
- In class we talked about the parametric description , ,
for the ellipse
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 isn't because Maple is stupid, but because this integral
really 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. In class
we graphed this for and
. Use the ParamPlot command to test the following
properties of this family.
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.
- 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
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William W. Farr