Subsections

# Parametric curves in the plane

## Purpose

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

## Getting Started

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

## Background

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 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.

### Derivatives and the slope

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 from class.

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.

### Arc length

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 is given below.

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.

## Exercises

1. 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 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.

2. 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.

3. 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.

4. Consider the family of parametric curves and , 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.
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.