Math 487 Lab 2

Part I. Apollonian Circles (as introduced in Ogilvy, pp. 13-17)

A. Viewing the circle as a trace

• Given two points and a number k >0, by definition the Apollonian locus that turns out to be an Apollonian circle is the set of points P with |AP|/|BP| = k.  How can be model this with Sketchpad?
• Here is a hint.  Given a segment PQ, let R be a point on the segment.  Hide the segment so that P, Q and R are showing, and construct segments PR and RQ.  Measure the ratio PR/QR.  If you leave P fixed and drag Q to change length PQ, then observe the ratio stays the same.
• Now, construct a circle with center A and radius PR and another circle with center B and radius QR.  Then when you drag Q, the intersection points S and T of the two circles will trace out the locus for a fixed k.

B. Constructing the circle using angle bisectors

• In a new sketch, given points A and B and a point C, construct the interior and exterior angle bisectors of angle C in triangle ABC and intersect these bisectors with line AB to give D and E.  Then the circle c1 with diameter DE passes through C and is an Apollonian circle.
• Make your Apollonian circle c1 red.  Once you have constructed the circle, trace the circle as you drag C to get a figure something like Ogilvy, page 18, or GTC p. 168.
• Now construct the circle c2 through A, B and C and color it blue.  Also trace this circle c2.  Drag C to get a figure something like Ogilvy, page 21, or GTC, page 170.
• Check by measuring the appropriate angle that the red circle and the blue circle are always orthogonal.

SAVE THIS FIGURE. You will return to it at the end of the lab.

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 lab02.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 lab02.gsp.

C. Getting Experience with Inversion Images

We will carry out Investigation 1 of Chapter 10, Exp 10.1 using the ready-made file lab02.gsp first. 

• In the lab02.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 lab02.gsp page Inv Path Tool. This is a GSP v 4 version of Investigation 2.

D. 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.

E. Inversive Geometry

• Read GTC 10.5, pp. 195-6 for basic definitions of inversive geometry.
• Carry out Investigation 1 of 10.5.  Use the tool Invert Path as Locus.
• Carry out Investigation 2 of 10.5.  Use the tool Invert Path as Locus.

F. Inversions of orthogonal pencils to "standard form"

Carry out Investigation 3, but read these new directions. Since in GSP v4 loci can be traced, the construction is much easier.

• The construction in Inv. 3, in the first figure on page 200, is the same as the last figure in Part B. Take this figure (make a new page as a copy if you like) and add a circle d with center A through a new point R. Make the appearance thick.
• Then use the tool Invert Path as Locus to invert in d the circles c1 and c2. Color c1 and its locus red and the other circle and its locus blue.
• In Part A, you already traced c1 and c2 as C moves. Turn off tracing for c1 and c2 but now trace the two loci. What does each trace look like?
• There appears to be a special point for the locus traces. Construct B' as the inversion of the point B in circle d. Is this the special point?
• Explain why the traces of the loci are what they appear to be and why they are related to B'. Why?

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 construct the elliptic pencil of circles and the orthogonal hyperbolic pencil of circles from two points A and B.
• Explain how to invert the two pencils above to get the set of lines through a point and the circles centered at the same point.
• 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. (We count the point at infinity as a point on the line to get the complete circle.)