Math 487 Lab 7: Images under inversion and Dr. Whatif's Euclidean Geometry

Part 1. Exploring images under inversion.

In the Lab 7 folder on the server are some sketches that illustrate images under inversion and some scripts that construct inversions. Spend 10 or 15 minutes exploring what the images of common figures look like. For example, what is the image of a triangle in various positions. What is the image of two lines intersecting at point P along with a circle centered at P. Remember that if you invert an image the image of the image is the original figure.

Part 2. Dr. Whatif's Euclidean Geometry. (DWEG)

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. This is the point that we are going to remove.

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

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

The angles between lines in DWEG are the usual angles between circles or between lines and circles in Euclidean 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 Euclidean 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. Make a script 'DW-line AB' that takes as givens, the points O, A, B.
2. Given a DWEG-line m through A and B and a DWEG-point C, construct a DWEG-line n through C which is perpendicular to m. (Note two possible cases: C is on m and C is not on m. Can you handle both with one sketch?) Make a script 'DW-perp AB thru C' that takes as givens, the points O, A, B, C.
3. Given a DWEG-line m through A and B and a DWEG-point C not on m, construct a DWEG-line p through C which is parallel to m. Make a script 'DW-parallel AB thru C' that takes as givens, the points O, A, B, C.
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 as circle inversion in Euclidean geometry.

First, carry out this construction in ordinary Euclidean geometry: take a 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. Thus B and all its images lie on a circle with center A. Construct the circle through B, B' and B'' and see that the other reflection images also lie on this circle.

Now we can mirror this circle construction in DWEG. Here is how we can find what a circle looks like. In DWEG take a 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 should give points so that the segments AB, AB', AB'', AB''', etc. are all congruent (if we choose to measure distance so that reflection is an isometry).

Now we make this important observation from Euclidean geometry. All these DWEG points B, B', B'', B'''' ... 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.
• Use this idea to construct a script 'DW-circle AB' with center A through B. Note: This circle will be a Euclidean circle, but A is the center in the DWEG geometry, not the usual Euclidean center.

DWEG Lab Activity. A Ruler made from Equal Steps.

First do this with a regular Euclidean figure.

• Let A and B be points on line 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. These points lie on p and make equally spaced ruler 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.

DWEG Lab Activity. A Triangle and a Square

Since you have made scripts that draw lines and circles, you have the Euclidean construction tools. Use these tools to construct these figures.

• Given any two DWEG points A and B, construct a DWEG equilateral triangle with side AB.
• Given any two DWEG points A and B, construct a DWEG square with side AB.