NAMES:

Load these packages:

> map(with,{plots,plottools,linalg}):

Warning, new definition for norm

Warning, new definition for trace

One afternoon, Loop Skywaltzer was out flying about in his red rocket scooter. Unfortunately, he forgot to turn off his laser, which shoots straight ahead (along the tangent line) from the nose cone of his scooter. There was a large space board (like bill boards, but these float in space) located on the x-z plane (also known as y = 0). At first, the company with an ad on this space board (Dunkin Danishes), was quite upset when they heard that Loop had burned a path through their space board. However, after seeing the results, they simply added a big cinnamon roll to their ad, and let Loop off the hook. Your job is to produce a graph (2 dimensional, since it lives in the x-z plane) of the path Loop burned out on their board. You should fully explain how you derived this plot however! For , Loop's coordinates were given by:

> loop:=t->[2+2*t+2*cos(3*t), 10-2*t, 22-5*t+3*sin(3*t)];

Here's a tubeplot of his path:

> tubeplot(loop(t),t=0..5,radius=.2,axes=normal,
labels=['x','y','z'],title="Loop's Path");

Click on the plot above (in a Maple window), and drag it around to get a feel for the "direction" he was flying.

>

Shown below is a plot (from Mathematica) with Loop's path and a few of the laser beams (red arrows). Note that the beams cut through the x-z plane (shown in tan).

Here's a hint at one way to proceed. I'll do this for a specific time, say t = 2. You'll have to do something similar for an unknown time t. Here is Loop's position when t = 2:

> loc:=loop(2.);

And the direction of the tangent line at this point:

> deriv:=diff(loop(t),t);

> dir:=eval(deriv,t=2.);

Thus, the y coordinate of the tangent line equation (for t = 2 anyway) is:

> y2:=s->6-2*s;

What s value gives y2 = 0?

> ans:=solve(y2(s)=0,s);

When t = 2, Loop's laser will cut the x-z plane at the point:

> cutpt:=evalf(loop(2)+3*dir);

Visual support. The dot is one point on the path which Loop burned in the space board!

> display([tubeplot(loop(t),t=0..2.5,radius=.2,
color=blue),
sphere(cutpt,.5),
line(loop(2.),loop(2.)+4*dir,color=red,thickness=3),
line([cutpt[1],0,0],cutpt,color=green),
line(cutpt,[0,0,cutpt[3]],color=green),
polygon([[0,0,0],[20,0,0],[20,0,25],[0,0,25]],
color=yellow,style=patch)],
axes=normal,labels=['x','y','z'],orientation=[42,54]);

Again, to get a better look, drag the plot around a bit. Good Luck!

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