Maple Project I

For this project you will create an animation showing the unit tangent and unit normal vectors following a curve such as displayed here.

There are a number of ways to accomplish this. Below I provide some general hints but I expect you to research more thoroughly using the maple resource links on the course web page or other web resources. You may also consult with each other but I expect each project to be an individual accomplishment.

This is a two part project. For the first part you will supply a maple worksheet (or document) with the commands for the animation. In the second part you will explain what each of the commands you used in Part I does. For this break out each command and explain what it does. A Microsoft Word document will suffice for the latter.

The due date for this will be March 30, 2006.

This project is for extra credit. You may receive from 0 to 5 points for completing it, 0-2 points for Part I and 0-3 points for Part II. These points will be added to your final grade.

Ok… Now let's get down to business.

First an important note: The following functions from the VectorCalculus library should not be used if you expect to get points for this project - TNBFrame, Tangent, TangentLine, TangentVector, PrincipalNormal.

We start with our usual 3 dimensional space curve, r(t) = x(t)i + y(t)j + z(t)k. We will use r(t) = cos(t)i + sin(t)j + t

The first thing you need to do is generate the Unit Tangent Vector, T(t) which, as you recall, is simply T(t) = r'(t)/||r'(t)||. Use the Linear Algebra or the Vector Calculus library to create a Maple expression or function for T(t).

Do the same for the Principal Unit Normal Vector,  N(t) which is N(t) = T'(t)/||T'(t)||. Again, create a Maple expression or function for N(t).

Now you can start plotting. First plot r(t) using the spacecurve(..) function.

Next you want to create arrows for T and N to be placed on the graph of r. Each arrow is created separately. You can use the arrow(..) function for this. There are several version of this function; one in the plots library and one in the plottools library. Choose as appropriate for your approach to the problem. Read up on the arrow function before choosing.

When you use the arrow function the first two parameters are the starting point and a vector to which the arrow is parallel. The starting point would be a point on the curve  r(t). The second parameter would be either T(t) or N(t), whichever is appropriate. Of course arrow expects these to resolve to an actual point and a particular vector so you must supply a value for t to make the function work.

Play around with the arrow function and use the display(..) function to display it to see how the arrow looks.

After you have an understanding of the arrow function you should begin animating. Remember that when you animate you sequence through a series of 'snapshots' in order to visualize movement. Look at the seq(..) function to create a sequence of arrow plots that you can then display.

When you are comfortable with this you may combine the original plot of r(t) with the two arrow sequences to generate the animated display.

Experiment with other space curves,  r(t) = x(t)i + y(t)j + z(t)k.