Math 487 Lab 3

Part I. Radical Axis Again

For a bit more about the computing the power function and finding the radical axis:

Download the file lab03-color-radaxis.gsp.

Part II. Images Under Inversion and Inversive Geometry

Setup

This part will be based on GTC Chapter 10. Some of the directions will be changed to work better with version 4 of GCP. In this lab we will use some inversion tools that can be taken from the previous lab or will be downloaded from links to the file lab03.gsp on this page. Also, as part of the download, you will find some GSP 4 versions of some of the sketches on the GTC disk, Chapter 10.

Click to download lab03.gsp.

A. Getting Experience with Inversion Images

We will carry out Investigation 1 of Chapter 10, Exp 10.1 using the ready-made file lab03.gsp first. 

• In the lab03.gsp file, look at the first two pages called Face 1 and Face 2. Drag points around and get a sense of what the inversion image of a figure looks like.
• Next, study the next page -- 1.1 Shape Inv. -- for each of the shapes. The goal is to understand what the image of each shape looks like. Pay particular attention to special points and special cases: for example, if the shape intersects the mirror circle, what is the image of the intersection points; if the shape passes through the center O of the mirror circle, what happens to the image?
• Now make some of these inverted shapes yourself. Go to the page Inv 1 and use the tool Invert Path as Locus to investigate further to be able to answer Question 1 of Invest. 1.
• Use the tool above to study the image of two perpendicular lines outside the circle. Then move them so that they intersect the circle. What is the image of the figure made of these two lines?
• Write down answers to all the parts of Question 1 at the end of Inv. 1.
• Make rough drawings to answer Question 2 at the end of Inv 1 and describe the special features of your drawings..
• Learn how the tool Invert Path as Locus was made by studying the lab03.gsp page Inv Path Tool. This is a GSP v 4 version of Investigation 2.

B. Inversion of a Line

• Carry out Exploration 10.2, Investigation 1. 
• Carry out and study the proof in Investigation 2.
• You should learn this proof.  Check the figure for whether the proof still works if the line intersects the mirror circle.

Note: Exploration 10.3 will be skipped in this lab.  These pages contain a proof that a circle not through the center of inversion inverts to another circle. This will be done in class.

C. Inversion of a Circle

• Carry out Exploration 10.3, Investigation 1. 
• Carry out Exploration 10.3, Investigation 2. Try to follow the explanation but then compare with the next sketch. Download the circle image sketch.
• In the circle image sketch, let p be the power of O with respect to circle e and let r be the radius of circle c. Then what is OA * OB and what is OA*OA'? Use your answers to conclude that OB/OA' is a constant for any position of circle e (not passing through O). Why does this prove that the inversion image of e is a circle?

D. Inversive Geometry

• As time permits, read and carry out Exploration 10.4, Investigations 1 and 2.
• Read over the tables and figures in Exploration 10.5 BEFORE Inv 1.

What you should carry away from this lab (now or after reflection)

• Be able to describe carefully, sketch or construct an inversion image of any segment, line, circle, arc, or figure made up of such objects.
• Be able to use conformality for more precise understanding of images.
• Understand and be able to write out the proof that the inversion image of a line (not through the center of inversion) is a circle.
• Understand the setup in section C that explains why the image of a circle is a circle (except for special case).