Introduction

In this short lab project, you will use Maple to visualize space curves, motion in space, and to understand the difference between a space curve and a position function .

Describing motion in space

To describe the motion of a particle moving in three-dimensional space, we must specify its position at each moment in time. This is a perfect use for a vector-valued function , i.e., a function of the form

where x , y , and z are all functions of t. We think of the variable t as "time" and of the vector P(t) as the "position at time t". We could visualize this vector as an arrow starting at the origin and ending at the point ( x (t), y (t), z (t)) where the particle lies at time t, but we usually just think of the point and ignore the vector. When used to describe the motion of a particle in space, a vector valued function is called a position function .

Space curves

Imagine now that our moving particle leaves a trail in space, something like the vapor trail left by a jet airplane. This trail is what we call the space curve parameterized by the vector valued function P (t). The variable t is a parameter , so this is simply a way of saying that the function P (t) is a way of describing the space curve as it elapses in time. More concretely, your signature on a piece of paper is a space curve, but the position of the point of the pen as a function of time is a position function.

Let's get started using Maple:

Open a Maple worksheet.

Type:

> with(plots);

The output from this command is a list of all of the special graphics commands contained in the "plots" package. When you start Maple, it loads only a small collection of general purpose commands (such as those we used last week). This helps to conserve memory and allows you to load just the commands you plan to use. If you would rather not see the list of plot commands, then you can use a colon instead of a semicolon at the end of the command line.

> with(plots):

You will notice "spacecurve" in the list of new commands in the "plots" package; this is the command we will use in this lab.

As above, we will first define a curve by using a vector-valued function named P . Because the symbols "<" and ">" are used for "less than" and "greater than", Maple uses square brackets "[" and "]" to define vectors. We want to examine the function P defined by:

> P := t -> [ t^2, t^3-t, cos(t) ];

The "spacecurve" command will plot our curve, but as with the "plot" command that plots graphs of functions of a single variable, we must specify a range of values for the variable. For example, if we want to plot P (t) as t ranges from -2 up to 2, we would input:

> spacecurve( P(t), t=-2..2);

If you are running Maple in the SEO 200 lab, your plot will appear in a separate window from your Maple worksheet. The plot window has controls for altering the appearance of the plot. Although the plot is three-dimensional, the two-dimensional monitor can only display a projection of the plot at any given moment. However, by clicking and dragging the plot you can change your perspective and view the plot from a different angle. After rotating the plot you must click the middle mouse button to redraw it from the new point of view. Try this now.

There are lots of other possibilities for altering the appearance of a plot; the following exercise will help you experiment with the different possibilities.

Exercise. Rotate the picture around and change the settings in the plot window to answer the following questions. (This exercise is purely for your own edification.)

For viewing space curves, which type of axes do you prefer: normal, box, frame, or none?

What is the difference between a "constrained" plot and an "unconstrained" plot?

What is the difference between "XY" coloring and "Z" coloring? Which would you use if you wanted to be able to tell whether two points on the curve had the same x- and y-coordinates? What if you wanted to be able to tell whether two points had the same z-coordinate?

Does the space curve that you just plotted cross itself?

You might want to search the on-line help pages for the topic "plotdevice" to learn about other ways in which Maple can present plots.

Seeing the path

We started out by using a vector-valued function to describe the motion of a particle through space, but the space curve we have just been looking at is a static object, i.e., it doesn't contain any information about when the particle occupied a particular point in space, just a record of the sum total of the locations that the particle occupied. The space curve also contains no information about the relative speeds of the particle at different moments in time. Is the particle moving quickly or slowly? Is the particle moving forward or backward or back and forth along the curve?

We can take advantage of the way that Maple plots space curves to visually recover some of this lost information. The way Maple plots a space curve on an interval is the same as the way it plots the graph of a function of one variable on an interval:

Maple breaks the interval into pieces of equal length.

Maple plots points for the value of the function at the endpoints of each segment.

Maple connects the plotted points with straight line segments.

Provided that Maple uses enough points (usually, something in the neighborhood of 20 will do), the curve looks smooth enough to fool our eyes. We can, however, ask Maple to plot just the points and not connect the dots, as it were, by using the "point" plot style. You can do this in two ways; the first way is to use the plot controls to set the plot style to "point", and the other way is by using the command option "style=POINT".

> spacecurve( P(t), t=-2..2, style=POINT);

Do this now, for yourself, using the plot controls to alter the plot style of the last plot you made. In terms of the airplane and vapor trail metaphore, the "point" plot style would be the result if the airplane dropped little balloons at regular intervals of time.

Before continuing, answer the following question for yourself: If the airplane is moving rapidly, will the balloons be closer together or further apart?

Exercise.

Use the "point" style to determine where the particle is moving slowly and where the particle is moving quickly.

Use Maple's help facility ("? spacecurve") to find out how many equally-spaced t-values Maple normally uses when plotting space curves. What happens to the plot if you increase the setting of this parameter to 40? (This is one parameter that can not be altered using the controls on the plot window.)

NOTE: There are many plot options and parameters that can not be altered with the controls on the plot window, and you can read about them with the Maple command:

> ? plot[options]

Here is an illustrative example in which we put labels on the x-, y-, and z-axes, color the curve red (Maple likes its colors in all caps), and use normal axes in the plot:

> spacecurve( P(t), t=-2..2, color=RED, axes=normal, labels=[x,y,z] );

Lab Assignment # 2

You are only expected to turn in the requested written answers.

1. Open a new Maple worksheet and define a vector-valued function that sends t to [ cos(t), sin(t), t ] . Name this function P .

2. Use Maple's spacecurve command to plot the space curve parameterized by P as t ranges from -2*Pi to 2*Pi . Write down a brief description of what the curve looks like.

3. Use the "labels" option (as above) to label the axes x , y , and z . Adjust your viewpoint so that the positive z -axis seems to be pointing straight out of the screen at you. (It should look like a point.) Write down a brief description of what you see.

4. Repeat Question 3 with the x -axis and y -axis. Explain how you might have been able to predict the results knowing only the definition of P .

5. Define a new vector-valued function P2 by replacing t with 2*t in the definition of P , i.e., define P2 to be the position function that takes t to [cos(2*t), sin(2*t), 2*t] . Plot the space curve parameterized by P2 from t = -2*Pi to t=2*Pi . Use the plot controls to change the plot style to "points" and the scaling to "constrained". Which path, P or P2 has greater speed?

6. Plot the space curve parameterized by P as t ranges from -4 to 4 . Now, define a new position function Q that maps t to [cos(t^2), sin(t^2), t^2 ] and plot the curve as t ranges from -2 to 2 . How do these two curves differ? By looking at your plots, what can you say about the velocities of the particles whose motion are described by P and Q near the point t=0 ? (Hint: P (0)= Q (0)=(1,0,0) . ) How do the velocities of the particles change over time?