Math 487 W98 Lab 3

This lab about inversion, especially about inversion as a transformation and the images of various objects.

You can choose the pace you want. The goal is to tie these experiments to what you are learning in class and from the reading, so pick a pace where you feel you are learning and making connections. When you feel you have gotten the point of the experiment, you should move on the next. But the goal should be to use the lab time the most effective way possible

If you do not finish, and if you feel the lab activity gives you valuable insights, you can continue them on your own later.

(If you finish early, there are several other experiments suggested that may help your understanding.)

Some of these constructions may help you when you do Assignment 5.

Note on Scripts and Script Tools

In some of the experiments, you are instructed to use a pre-written script. All these scripts can be found in the folder Circle Tools in the Math 487 Class Folder. These work much better if you install them as Script Tools. You can do this by making the Circle Tools folder your script tools folder. Ask how if you don't know how.

Experiment 1. Free exploration of images.

This is designed as a free exploration to help you build some intuition about inversion. It could go on and on, but probably 30-40 minutes is enough for the time being. Do as much as you comfortably can of Exp 10.1, Inv1, on pp 179-182 of GTC. You may find Inv. 2 also valuable for the homework if you have time later.

Experiment 2. Inversion of Lines.

Do GTC 10.2, Investigation 1, pp. 187-89. Do this exploration.
Then make the construction in Investigation 2. This proof was presented in class Wed. morning, so you can explore the corresponding figure here.

Experiment 3. Inversion of Circles

Do GTC 10.3, Investigation 1, pp. 190-192. Be sure to get a good feeling for what the image of a circle d looks like when the circle d is inside, outside, or intersecting the mirror circle. Also be sure to examine the case whe the center of the mirror circle is inside the circle d.
Investigation 2 includes one proof of the theorem on images of circles. Sved has another proof. The proof in Ogilvy is closer to the one in GTC. You may want to look at this now or defer it until later at a quieter time.

Experiment 4. Figures in the Inversive Plane

Do GTC 10.5, Investigations 1 and 2, pp. 195-199. Note: The inversive plane is the usual plane with the point Infinity added to it.

Additional Experiments.

For understanding families of orthogonal circles, GTC 10.5, Inv 3 and the preceding 9.4 are very helpful. You may want to use these to help study these topics.