Math 487 Lab 4

Part 1. Quick Look at Apollonian Circles

Explore Briefly the two sketches on Apollonian Circles in the Lab 4 Folder in the Class Folders.

Part 2. (The Main Event) Dr. Whatif's Euclidean Geometry.

This is an example of a model for Euclidean geometry which is not the standard (x,y) plane. We call this DWEG, for Dr. Whatif's Euclidean Geometry.

Here is the model: Choose a point in the ordinary plane and label it O.

The points in DWEG are the points of the plane excluding O but including the point at infinity.

The lines in DWEG are ordinary circles or lines through O. Since O is the point that is not there, it is not included as a point in any DWEG-line. But the point at infinity is considered one of the points of a DWEG-line defined by a Euclidean line.

The angles between lines in DEWG are the usual angles between circles or between lines and circles in Eucidean geometry.

DWEG Lab Activity. The Basics.

Now here is the drill. Use what you know about constructing circles to carry out the basic constructions of Eudlidean geometry in this model.

1. Given two DWEG-points A and B construct a DWEG-line through A and B (in other words, construct a circle; you can set aside the special case when the DWEG-line is a Euclidean line.
2. Given a DWEG-line m and a DWEG-point A, construct a DWEG-line n through A which is perpendicular to m. (Note two possible cases: A is on m and A is not on m. Can you handle both with one sketch?)
3. Given a DWEG-line m and a DWEG-point A not on m, construct a DWEG-line p through A which is parallel to m.
4. Construct a rectangle in DWEG by constructing two parallel lines and two lines perpendicular to the parallels.

DWEG Lab Activity. Circles.

We don't yet know how to measure distance in DWEG, but we do know how to reflect across a line. We declare line reflection in DWEG to be the same a circle inversion in Euclidean geometry.

Now, here is how we can find what a circle looks like. Take a DWEG point A and two lines m and n through A. Now choose any point B. Reflect B across m to get B', then reflect B' across n to get B'' then reflect B'' across m to get B''', etc. This gives points so that the segments AB, AB', AB'', AB''', etc. are all congruent (if we believe that reflection is an isometry.

All these points lie on a Euclidean circle c. (You can see that this looks correct with Sketchpad, why is it true?

• So construct the Euclidean circle c through B, B' and B'' to get the circle in DWEG.

DWEG Lab Activity. A Ruler made from Equal Steps.

First do this with a regular Euclidean figure.

• Let A and B be points on p. Construct lines a and b through A and B which are perpendicular to p.
• Now mark equal steps on p, reflect A across b to get A' and then reflect A' across a and then continue reflecting across b and then a over and over.
• Also do this for B. Look for the rule markings on p.
• Remember the link between double reflection and translation.

Now do the same for a DWEG figure to construct a ruler on a DWEG line p.